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+{"text":"\\section{Introduction}\n\nThe mathematical challenge associated with the position-dependent mass (PDM)\nvon Roos Hamiltonian \\cite{1}, and the feasible applicability of the PDM\nsettings in different fields of physics, has inspired a relatively intensive\nrecent research attention on the quantum mechanical (see the sample of\nreferences \\cite{2,3,4,5,6,7,8,9,10,11,12}), classical mechanical and\nmathematical (see the sample of references \\cit\n{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37\n) domains in general. The position-dependent mass is, in principle, a\nposition-dependent deformation in the standard constant mass settings that\nintroduces it's own PDM-byproducted reaction-type force $R_{PDM}\\left( x\n\\dot{x}\\right) =m^{^{\\prime }}\\left( x\\right) \\dot{x}^{2}\/2$, and manifests\ndeformation in the potential force field that may inspire nonlocal\nspace-time point transformations. That is, if a PDM-particle is moving in a\nharmonic oscillator potential force field $V\\left( x\\right) =m\\left(\nx\\right) \\omega ^{2}x^{2}\/2$, for example, then one may use $q=x\\sqrt\nm\\left( x\\right) }\\Longrightarrow V\\left( q\\right) =\\omega ^{2}q^{2}\/2$ to\nretain the standard constant mass settings. In the process, some\nposition-dependent deformation in time may be deemed vital (see \\cite{22,38}\nfor more details on this issue). For some comprehensive discussions on the\nquantum mechanical PDM related ordering ambiguity and on the\nclassical-quantum mechanical correspondence, the reader may refer to (c.f.,,\ne.g., \\cite{12,17}). However, on the issue of the classical mechanical\nequivalence between the Euler-Lagrange's and Newton's equations of motion\none may refer to \\cite{22}.\n\nIn a very recent study, Mustafa \\cite{38} has introduced a general nonlocal\npoint transformation for one-dimensional PDM Lagrangians and provided their\nmappings into a constant \\emph{\"unit-mass\"} Lagrangians in the generalized\ncoordinates. Therein, it has been shown that the applicability of such\nmappings not only results in the linearization of some nonlinear oscillators\nbut also extends into the extraction of exact solutions of more complicated\ndynamical systems. Hereby, the exactly solvable Lagrangians (labeled as \n\\emph{\"reference\/target-Lagrangians\"}) are mapped along with their exact\nsolutions into PDM-Lagrangians (labeled as \\emph\n\"target\/reference-Lagrangians\"}). It would be natural and interesting to\nextend\/generalize Mustafa's methodical proposal \\cite{38} to deal with\nLagrangians in more than one-dimension. Therefore, the current methodical\nproposal is a parallel extension to \\cite{38} and deals with PDM Lagrangians\nin two-dimensions.\n\nHowever, in handling such higher dimensional Lagrangians the notion\/concept\nof \\emph{superintegrability} (c.f., e.g., \\cit\n{39,40,41,42,43,44,45,46,47,48,49,50,51} and related references cited\ntherein) is unavoidable. A Lagrangian system is said to be \\emph\nsuperintegrable} if it admits the Liouville-Arnold sense of integrability\nand introduces more constants of motion (also called integrals of motion)\nthan the degrees of freedom the system is moving within (c.f., e.g., \\cit\n{45}). The set of the two-dimensional isotonic oscillator potential\n\\begin{equation}\nV(x_{1},x_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}x_{1}^{2}+\\omega\n_{2}^{2}x_{2}^{2}+\\frac{\\beta _{1}}{x_{1}^{2}}+\\frac{\\beta _{2}}{x_{2}^{2}\n\\right) ,\n\\end{equation\nfor example (a more general extension of the Smorodinsky-Winternitz isotonic\noscillator potentials where $\\ \\omega _{1}=\\omega _{2}=\\omega $) is known to\nbe \\emph{superintegrable} (c.f., e.g., \\cite{39,40,41,42,46,47}).\\ The\ndetails on their \\emph{superintegrability} (c.f.,e.g., \\cite{39,41}) or\ntheir \\emph{maximal superintegrability} (c.f., e.g., \\cite{40})\nclassification criteria lay far beyond the scope of our study here. They are\nconsidered as \\emph{superintegrable} potentials throughout the current\nmethodical proposal, though we verify their \\emph{superintegrability} in\nbrief to make the current methodical proposal self-contained. On the other\nhand, under some nonlocal space-time transformation (see (10) below) the\ntwo-dimensional \\emph{superintegrable} isotonic oscillator potential\n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}q_{1}^{2}+\\omega\n_{2}^{2}q_{2}^{2}+\\frac{\\beta _{1}}{q_{1}^{2}}+\\frac{\\beta _{2}}{q_{2}^{2}\n\\right) ,\n\\end{equation\n(where $q_{j}\\equiv q_{j}\\left( x_{j}\\right) \\,;j=1,2$ are some invertible, \n\\partial x_{j}\/\\partial q_{j}\\neq 0\\neq \\partial q_{j}\/\\partial x_{j}$,\ngeneralized coordinates) may yield a PDM-deformed isotonic oscillator\npotentials. Hereby,\\ we argue that, if the exactly solvable \\emph\n\"superintegrable reference-Lagrangians\"} are mapped into\\ \\emph{\"PDM\ntarget-Lagrangians\"}, then the set of \\emph{\"PDM target-Lagrangians\"} is a\nset of \\emph{\"sub-superintegrable PDM-Lagrangians\"} (as shall be so labeled\nhereinafter). In addition to our main objective to extend\/generalize\nMustafa's methodical proposal \\cite{38} to deal with two-dimensional\nLagrangians, we anticipate that the introduction of the terminology of \\emph\n\"sub-superintegrability\"} \\ would\\ yet add a new flavour\/concept to \\emph\n\"superintegrability\"}. The organization of our methodical proposal is in\norder.\n\nThe two-dimensional nonlocal space-time PDM transformation and the\nEuler-Lagrange equations' invariance are discussed in section 2. Therein,\none would observe that the obvious non-separability of the two-dimensional \n\\emph{\"PDM target-Lagrangians\"} $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $ is accompanied by the separability of the two-dimensional \n\\emph{\"reference-Lagrangians\"} $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $. Mandating in effect \\emph{\"sub-separability\"} of the\ntwo-dimensional \\emph{\"PDM target-Lagrangians\"} $L\\left( x_{1},x_{2},\\dot{x\n_{1},\\dot{x}_{2};t\\right) $, as a result of the two-dimensional nonlocal PDM\ntransformations. In section 3, the mapping from the \\emph{superintegrability}\nof some linear oscillators into \\emph{sub-superintegrability} of nonlinear\nPDM-oscillators is introduced. To make the current methodical proposal\nself-contained, we discuss in short the superintegrability of the \\emph\nreference-Lagrangians} (although similar discussions are available in the\nliterature). Three two-dimensional \\emph{sub-superintegrable}\nMathews-Lakshmanan type PDM-oscillators are used for clarification. They\nare, (i) a \\emph{sub-superintegrable} Mathews-Lakshmanan type-I\nPDM-oscillator for a PDM-particle moving in a harmonic oscillator potential,\n(ii) a \\emph{sub-superintegrable} Mathews-Lakshmanan type-II PDM-oscillator\nfor a PDM-particle moving in a constant potential, (iii) a \\emph\nsub-superintegrable} shifted Mathews-Lakshmanan type-III PDM-oscillator for\na PDM-particle moving in a a shifted harmonic oscillator potential. In the\nsame section, we discuss (in short) the superintegrability of some shifted\nlinear oscillators (a new superintegrable model to the best of our\nknowledge) and report their mapping into sub-superintegrable nonlinear\nPDM-Oscillators. We use, moreover, a \\emph{superintegrable} isotonic\noscillator and map it into a \\emph{sub-superintegrable} PDM isotonic\noscillator. Of course the mappings also include exact solvability of the \n\\emph{\"reference-Lagrangians\"} at hand as well. Our concluding remarks are\ngiven in section 4.\n\n\\section{Two-dimensional nonlocal PDM-point transformations and\nEuler-Lagrange's invariance}\n\nThe Lagrangian of a particle with a constant \\emph{\"unit mass\"} moving in\nthe generalized coordinates $\\left( q_{1},q_{2}\\right) \\equiv \\left(\nq_{1}\\left( x_{1}\\right) ,q_{2}\\left( x_{2}\\right) \\right) $, under the\ninfluence of a potential force field $V(q_{1},q_{2}),$ and a\ndeformed\/re-scaled time $\\tau $, is given b\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -V(q_{1},q_{2})\\text{ }\n\\text{ \\ }\\tilde{q}_{j}=\\frac{dq_{j}}{d\\tau };\\,j=1,2.\n\\end{equation\nThe\\ corresponding Euler-Lagrange equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\left( \\frac{\\partial L}{\\partial \\tilde{q}_{j}}\\right) \n\\frac{\\partial L}{\\partial q_{j}}=0;\\,\\,j=1,2,\n\\end{equation\ntherefore, read \n\\begin{equation}\n\\text{\\ }\\frac{d}{d\\tau }\\tilde{q}_{j}+\\frac{\\partial }{\\partial q_{j}\nV(q_{1},q_{2})=0;\\,j=1,2.\n\\end{equation\nOne should notice that for a \\emph{\"unit mass\"} particle moving in a \\emph\nfree} force field $V(q_{1},q_{2})=0\\Longrightarrow d\\tilde{q}_{j}\\left(\nx_{j}\\right) \/d\\tau =0$, hence the linear momenta $\\tilde{q}_{1}\\left(\nx_{1}\\right) $ and $\\tilde{q}_{2}\\left( x_{2}\\right) $ are conserved\nquantities ( in this particular case) and serve as fundamental integrals\n(i.e., constants of motion). However, for the set of potentials in the from\nof \n\\begin{equation}\nV(q_{1},q_{2})=V_{1}\\left( q_{1}\\right) +V_{2}\\left( q_{2}\\right) \\neq\n0;\\,V_{j}\\left( q_{j}\\right) \\neq 0;\\,j=1,2\n\\end{equation\n(which is the set of potential force fields of our interest in the current\nstudy) one may recast (5) a\n\\begin{equation}\n\\text{\\ }\\frac{d}{d\\tau }\\tilde{q}_{j}\\left( x_{j}\\right) +\\frac{\\partial }\n\\partial q_{j}}V_{j}(q_{j})=0\\Longrightarrow \\tilde{q}_{j}\\left( x\\right) \n\\text{\\ }\\frac{d}{d\\tau }\\tilde{q}_{j}\\left( x\\right) +\\tilde{q}_{j}\\left(\nx\\right) \\frac{\\partial }{\\partial q_{j}}V_{j}(q_{j})=0.\n\\end{equation\nto obtain two integrals of motion $I_{1}=E_{1}$ and $I_{2}=E_{2}$ via \n\\begin{equation}\n\\frac{d}{d\\tau }\\left[ \\frac{1}{2}\\tilde{q}_{j}^{2}+V_{j}(q_{j})\\right]\n=0\\Longrightarrow \\frac{dE_{j}}{d\\tau }=\\frac{dI_{j}}{d\\tau }=0;\\,j=1,2.\n\\end{equation\nYet, the conservation of the total energy $E_{tot}$ is a natural and an\nimmediate consequence of such settings. That is\n\\begin{equation}\n\\frac{d}{d\\tau }\\left[ E_{1}+E_{2}\\right] =\\frac{d}{d\\tau }\\left[ \\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) +V_{1}\\left( q_{1}\\right)\n+V_{2}\\left( q_{2}\\right) \\right] =0\\Longrightarrow \\frac{dE_{tot}}{d\\tau }=0\n\\end{equation\nThe separability and\/or integrability of the above system is obvious,\ntherefore.\n\nNext, let us introduce the nonlocal point transformation of the for\n\\begin{equation}\nq_{j}\\equiv q_{j}\\left( x_{j}\\right) =\\int \\sqrt{g\\left( \\bar{x}\\right) \ndx_{j}\\text{, \\ }\\tau =\\int f\\left( \\bar{x}\\right) dt\\Longrightarrow \\frac\nd\\tau }{dt}=f\\left( \\bar{x}\\right) \\neq 0\\text{ };\\text{ \\ }x_{j}\\equiv\nx_{j}\\left( t\\right) ,\\,\\bar{x}=x_{1},x_{2}.\n\\end{equation\nConsequently, with an overhead dot to identify total time $t$ derivative, \n\\begin{equation}\n\\frac{dq_{j}}{d\\tau }=\\tilde{q}_{j}=\\frac{\\dot{x}_{j}\\sqrt{g\\left( \\bar{x\n\\right) }}{f\\left( \\bar{x}\\right) }\\text{ },\\text{ \\ \\ \\ }\\frac{d}{d\\tau \n\\tilde{q}_{j}\\left( x_{j}\\right) =\\frac{\\sqrt{g\\left( \\bar{x}\\right) }}\nf\\left( \\bar{x}\\right) ^{2}}\\left[ \\ddot{x}_{j}+\\left( \\frac{1}{2}\\frac{\\dot\ng}\\left( \\bar{x}\\right) }{g\\left( \\bar{x}\\right) }-\\frac{\\dot{f}\\left( \\bar{\n}\\right) }{f\\left( \\bar{x}\\right) }\\right) \\dot{x}_{j}\\right] .\n\\end{equation\nWhich when substituted in (7) would , in a straightforward manner, result in \n\\begin{equation}\n\\left[ \\dot{x}_{1}\\,\\ddot{x}_{1}+\\dot{x}_{2}\\,\\ddot{x}_{2}+\\left( \\frac{1}{2\n\\frac{\\dot{g}\\left( \\bar{x}\\right) }{g\\left( \\bar{x}\\right) }-\\frac{\\dot{f\n\\left( \\bar{x}\\right) }{f\\left( \\bar{x}\\right) }\\right) \\left( \\dot{x\n_{1}^{2}+\\dot{x}_{2}^{2}\\right) \\right] +\\frac{f\\left( \\bar{x}\\right) ^{2}}\ng\\left( \\bar{x}\\right) }\\frac{d}{dt}V\\left( \\bar{x}\\right) =0;\\,x_{j}\\equiv\nx_{j}\\left( q_{j}\\right) ,\n\\end{equation\nwher\n\\begin{equation*}\n\\frac{d}{dt}V\\left( \\bar{x}\\right) =\\left( \\dot{x}_{1}\\frac{\\partial }\n\\partial x_{1}}V_{1}\\left( x_{1}\\right) +\\dot{x}_{2}\\frac{\\partial }\n\\partial x_{2}}V_{2}\\left( x_{2}\\right) \\right) .\n\\end{equation*}\n\nOn the other hand, for a two-dimensional position-dependent mass particle\nmoving in a force field $V\\left( x_{1},x_{2}\\right) =V_{1}\\left(\nx_{1}\\right) +V_{2}\\left( x_{2}\\right) $ the Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left( \n\\bar{x}\\right) \\left( \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}\\right) -\\left[\nV_{1}\\left( x_{1}\\right) +V_{2}\\left( x_{2}\\right) \\right] ;,\n\\end{equation\nin the Cartesian coordinates, would yield two Euler-Lagrange equations \n\\begin{equation}\nm\\left( \\bar{x}\\right) \\ddot{x}_{j}+\\dot{m}\\left( \\bar{x}\\right) \\dot{x}_{j}\n\\frac{1}{2}\\frac{\\partial m\\left( \\bar{x}\\right) }{\\partial x_{j}}\\left( \n\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}\\right) +\\frac{\\partial }{\\partial x_{j}\nV_{j}\\left( x_{j}\\right) =0;\\,j=1,2.\n\\end{equation\nThe non-separability of this system is obviously manifested by the\nposition-dependent mass term. However, when multiplied, from the left, by \n\\dot{x}_{j}$ it reads \n\\begin{equation}\nm\\left( \\bar{x}\\right) \\dot{x}_{j}\\ddot{x}_{j}+\\dot{m}\\left( \\bar{x}\\right) \n\\dot{x}_{j}^{2}-\\frac{1}{2}\\left( \\dot{x}_{j}\\frac{\\partial m\\left( \\bar{x\n\\right) }{\\partial x_{j}}\\right) \\left[ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2\n\\right] +\\left( \\dot{x}_{j}\\frac{\\partial }{\\partial x_{j}}\\right)\nV_{j}\\left( x_{j}\\right) =0;\\,j=1,2.\n\\end{equation\nConsequently, the addition of the two equations of (15) yield\n\\begin{equation}\n\\dot{x}_{1}\\ddot{x}_{1}+\\dot{x}_{2}\\ddot{x}_{2}+\\frac{1}{2}\\frac{\\dot{m\n\\left( \\bar{x}\\right) }{m\\left( \\bar{x}\\right) }\\left( \\dot{x}_{1}^{2}+\\dot{\n}_{2}^{2}\\right) +\\frac{1}{m\\left( \\bar{x}\\right) }\\frac{d}{dt}V\\left( \\bar{\n}\\right) =0,\n\\end{equation\nand henc\n\\begin{equation}\n\\frac{d}{dt}\\left[ \\frac{1}{2}m\\left( \\bar{x}\\right) \\,\\left( \\dot{x\n_{1}^{2}+\\dot{x}_{2}^{2}\\right) +V_{1}\\left( x_{1}\\right) +V_{2}\\left(\nx_{2}\\right) \\right] =0\\Longrightarrow \\frac{dE_{tot}}{dt}=0.\n\\end{equation\nHereabout, we emphasize that the conservation of the total energy $E_{tot}$\nin (17) can \\emph{never} be considered as an immediate consequence of the\nsum of two fundamental integrals of motion $E_{x_{1}}$ and $E_{x_{2}}$. The\ntime evolutions of $E_{x_{j}j}$'s do not satisfy (15). i.e., one may easily\nshow that \n\\begin{equation}\n\\frac{d}{dt}E_{x_{j}}=\\frac{d}{dt}\\left[ \\frac{1}{2}m\\left( \\bar{x}\\right) \n\\dot{x}_{j}^{2}+V_{j}\\left( x_{j}\\right) \\right] =m\\left( \\bar{x}\\right) \n\\dot{x}_{j}\\ddot{x}_{j}+\\frac{1}{2}\\dot{m}\\left( \\bar{x}\\right) \\dot{x\n_{j}^{2}+\\dot{x}_{j}\\frac{\\partial }{\\partial x_{j}}V_{j}\\left( x_{j}\\right)\n\\neq 0\n\\end{equation\ncompared to (15). The third term in (15) is missing in (18). Moreover, the\ncomparison between (12) and (16) obviously suggests that the Euler-Lagrange\nequations of motion (12) and (16) are identical if and only if $f\\left( \\bar\nx}\\right) $ and $g\\left( \\bar{x}\\right) $ satisfy the conditions \n\\begin{equation}\ng\\left( \\bar{x}\\right) =m\\left( \\bar{x}\\right) \\,f\\left( \\bar{x}\\right)\n^{2}\\Longleftrightarrow \\frac{1}{2}\\frac{\\dot{g}\\left( \\bar{x}\\right) }\ng\\left( \\bar{x}\\right) }-\\frac{\\dot{f}\\left( \\bar{x}\\right) }{f\\left( \\bar{x\n\\right) }=\\frac{1}{2}\\frac{\\dot{m}\\left( \\bar{x}\\right) }{m\\left( \\bar{x\n\\right) }\\Longleftrightarrow q_{j}=\\int \\sqrt{m\\left( \\bar{x}\\right) \nf\\left( \\bar{x}\\right) dx_{j}.\n\\end{equation\nAs such, it is clear that the functional nature\/structure of the\nposition-dependent mass $m\\left( \\bar{x}\\right) $ determines the\nnature\/structure of the nonlocal transformation functions $g\\left( \\bar{x\n\\right) $ and $\\,f\\left( \\bar{x}\\right) .$ For the sake of simplicity,\nhowever, we shall work with $m\\left( \\bar{x}\\right) \\equiv m\\left( r\\right)\n\\Longrightarrow g\\left( \\bar{x}\\right) \\equiv g\\left( r\\right) $ and \n\\,f\\left( \\bar{x}\\right) \\equiv $ $f\\left( r\\right) $. Hence, $m\\left(\nr\\right) $, $g\\left( r\\right) $ and $f\\left( r\\right) $ are well behaved\nfunctions of explicit dependence on $r=\\sqrt{x_{1}^{2}+x_{2}^{2}}$, if not\notherwise mentioned.\n\nAt this point, one may safely conclude that under such invertible (i.e., the\nJacobian determinant $det\\left( \\partial x_{i}\/\\partial q_{i}\\right) \\neq 0\n) nonlocal transformation, (10) and (19), the two-dimensional Euler-Lagrange\nequations, (12) and (16), remain invariant. That is, the Lagrangian $L\\left(\nq_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) ,$ in (3), non-locally\ntransforms (via (10) and (19)) into the PDM Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $, in (13), and leaves in the process the\ncorresponding two-dimensional Euler-Lagrange equations of motions, (12) and\n(16), invariant. The mappin\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right)\n\\Longleftrightarrow \\left\\{ \n\\begin{array}{c}\ng\\left( r\\right) =m\\left( r\\right) f\\left( r\\right) ^{2}\\medskip \\medskip \\\\ \nq_{j}=\\int \\sqrt{m\\left( r\\right) }f\\left( r\\right)\ndx_{j}\\,;\\,j=1,2\\smallskip \\medskip \\\\ \n\\tau =\\int f\\left( r\\right) dt\\medskip \\\\ \n\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) };\\,j=1,\n\\end{array\n\\right\\} \\Longleftrightarrow L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) ,\n\\end{equation\nbetween the \\emph{\"unit mass\" } Lagrangian $L\\left( q_{1},q_{2},\\tilde{q\n_{1},\\tilde{q}_{2};\\tau \\right) $ and PDM Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ is clear, therefore.\n\nNevertheless, the obvious non-separability of the two-dimensional\nEuler-Lagrange equations associated with the PDM Lagrangian $L\\left(\nx_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ (hence, $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ is non-separable) is accompanied by the\nseparability of the two-dimensional Euler-Lagrange equations associated with\nunit mass Lagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau\n\\right) $ (hence, $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau\n\\right) $ is separable). This should, in turn, mandate the notion of\n\"sub-separability\" of the two-dimensional Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ as a result of the two-dimensional PDM\nnonlocal transformations in (20). Likewise, if the two-dimensional unit mass\nLagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) $\nadmits superseparability and\/or superintegrability then the two-dimensional\nPDM Lagrangian $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ may\nvery well be labeled as sub-superseparable and\/or sub-superintegrable. The\nlatter forms the focal point of our study here and shall be clarified in the\nforthcoming experimental examples.\n\n\\section{From superintegrability to sub-superintegrability; two-dimensional\nPDM-oscillators}\n\nAlthough the superintegrability of some of the \\emph{reference\nsuperintegrable} oscillators, we use here, is verified in the literature, we\nrecycle them in such a way that serves our methodical proposal and keeps it\nself-contained. The\\emph{\\ superintegrable linear oscillators}, the \\emph\nsuperintegrable shifted-oscillators} (is a new \\emph{superintegrable}\noscillator model, to the best of our knowledge), and the \\emph\nsuperintegrable isotonic oscillators} are used here as illustrative\nexamples. They are mapped along with their exact solutions into \\emph\nsub-superintegrable PDM-oscillators}.\n\n\\subsection{Superintegrable linear oscillators into sub-superintegrable\nnonlinear PDM-oscillators}\n\nConsider a \\emph{unit mass }particle moving under the influence of the\ntwo-dimensional oscillators potential \n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}q_{1}^{2}+\\omega\n_{2}^{2}q_{2}^{2}\\right) \\Longrightarrow V_{j}\\left( q_{j}\\right) =\\frac{1}{\n}\\omega _{j}^{2}q_{j}^{2},\\,\\omega _{j}=n_{j}\\omega _{\\circ };\\,j=1,2,\n\\end{equation\nin the generalized coordinates. Then, the corresponding two-dimensional\nLagrangia\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -\\frac{1}{2}\\left( \\omega\n_{1}^{2}q_{1}^{2}+\\omega _{2}^{2}q_{2}^{2}\\right) ,\n\\end{equation\nleads to Euler-Lagrange equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\tilde{q}_{j}+\\omega _{j}^{2}q_{j}=0\\Longrightarrow\nq_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega _{j}\\tau +\\varphi \\right)\n,\\,\\,j=1,2,\n\\end{equation\nsubjected to the boundary conditions $\\,q_{j}\\left( 0\\right) =A_{j},\\,\\tilde\nq}_{j}\\left( 0\\right) =0$, say. It obviously admits separability and is\nknown to satisfy the superintegrability conditions (c.f. e.g., \\cit\n{39,40,41,42}) via the use of \\ a complex factorization technique (c.f.,\ne.g., \\cite{43,44}). That is, if we introduce the two complex functions \n\\begin{equation}\nQ_{j}=\\tilde{q}_{j}+i\\omega _{j}q_{j}\\Longrightarrow \\frac{d}{d\\tau \nQ_{j}=i\\omega _{j}Q_{j};\\,\\,j=1,2,\n\\end{equation\nthen the function\n\\begin{equation}\nQ_{jk}=Q_{j}^{\\omega _{k}}\\left( Q_{k}^{\\ast }\\right) ^{\\omega\n_{j}};\\,\\,j,k=1,2,\n\\end{equation\nrepresent \\emph{complex }constants of motion with vanishing\ndeformed\/rescaled-time evolution $\\tau $\n\\begin{equation}\n\\frac{d}{d\\tau }Q_{jk}=i\\left( \\omega _{j}\\omega _{k}-\\omega _{k}\\omega\n_{j}\\right) Q_{jk}=0.\n\\end{equation\nMoreover, one can, in a straightforward manner, verify that \n\\begin{equation}\nQ_{jj}=\\tilde{q}_{j}^{2}+\\omega _{j}^{2}q_{j}^{2}\\Longrightarrow\n2E_{1}=I_{1}=Q_{11},\\,2E_{2}=I_{2}=Q_{22},\n\\end{equation\nwhere $I_{1}$ and $I_{2}$ are two fundamental integrals of motion. Yet, for\nthe isotropic oscillator $\\omega _{1}=\\omega _{2}=\\omega _{\\circ }$, for\nexample\n\\begin{equation}\nQ_{12}=\\left( \\tilde{q}_{1}\\tilde{q}_{2}+\\omega _{\\circ\n}^{2}q_{1}q_{2}\\right) +i\\omega _{\\circ }\\left( q_{1}\\tilde{q}_{2}-q_{2\n\\tilde{q}_{1}\\right) =I_{3}+iI_{4}\n\\end{equation\nidentifies two more integrals of motion \n\\begin{equation}\nI_{3}=\\func{Re}Q_{12}=\\tilde{q}_{1}\\tilde{q}_{2}+\\omega _{\\circ\n}^{2}q_{1}q_{2}\\text{ \\ ; \\ \\ }I_{4}=\\func{Im}Q_{12}=\\omega _{\\circ }\\left(\nq_{1}\\tilde{q}_{2}-q_{2}\\tilde{q}_{1}\\right) .\n\\end{equation\nwhich are, in the general case $\\omega _{1}\\neq \\omega _{2}$, polynomials in\nthe momenta. Therefore, our two-dimensional Lagrangian (22) admits\nsuperintegrability in the generalized coordinates $\\left( q_{1},q_{2}\\right) \n$ and in the deformed\/rescaled time $\\tau $. The details on such\nsuperintegrability are far beyond the scope of our the current methodical\nproposal, though can be traced through the comprehensive article of Ranada \n\\cite{42} and related references cited therein.\n\n\\subsubsection{Sub-superintegrable Mathews-Lakshmanan type-I PDM-oscillators}\n\nLet us now consider a PDM particle $m\\left( r\\right) $\\ moving in the\nharmonic oscillator force field \n\\begin{equation*}\nV\\left( x_{1},x_{2}\\right) =\\frac{1}{2}m\\left( r\\right) \\omega ^{2}\\left(\nx_{1}^{2}+x_{2}^{2}\\right) ,\n\\end{equation*\nwith the corresponding PDM Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left(\nr\\right) \\left[ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left(\nx_{1}^{2}+x_{2}^{2}\\right) \\right] .\n\\end{equation\nThis Lagrangian indulges one and only one integral offered by the total\nenergy $E_{tot}$ as in (17), and the corresponding Euler-Lagrange equations\nare non-separable. However, with the substitutio\n\\begin{equation}\nq_{j}=x_{j}\\sqrt{m\\left( r\\right) }\\,;\\smallskip \\medskip \\,j=1,2,\n\\end{equation\none can non-locally transform $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $ of (30) into $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ of (22). Moreover, if this substitution is used along\nwith the transformation in (20) we ge\n\\begin{equation}\nq_{j}=x_{j}\\sqrt{m\\left( r\\right) }\\Longrightarrow \\frac{dq_{j}}{dx_{j}}\n\\sqrt{m\\left( r\\right) }\\left[ 1+\\frac{m^{\\prime }\\left( r\\right) }{2m\\left(\nr\\right) }\\left( \\frac{x_{j}^{2}}{r}\\right) \\right] ,\\medskip\n\\end{equation\nan\n\\begin{equation*}\nq_{j}=\\int \\sqrt{m\\left( r\\right) }f\\left( r\\right) dx_{j}\\Longrightarrow \n\\frac{dq_{1}}{dx_{1}}+\\frac{dq_{2}}{dx_{2}}=2\\sqrt{m\\left( r\\right) }f\\left(\nr\\right) .\n\\end{equation*\nhenc\n\\begin{equation}\nf\\left( r\\right) =1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left( r\\right) }{m\\left(\nr\\right) }r.\n\\end{equation\nAt this point, one should be aware that we are interested in $m\\left(\nr\\right) $ and $f\\left( r\\right) $ that are only explicit function in $r\n\\sqrt{x_{1}^{2}+x_{2}^{2}}$. Obviously, moreover, the choice of $f\\left(\nr\\right) $ would determine the position-dependent mass function $m\\left(\nr\\right) $ (of course the other way around works as well). This is clarified\nin the following assumption. Let us assume tha\n\\begin{equation}\nf\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\n\\end{equation\nto obtain a position-dependent mass of the for\n\\begin{equation}\nf\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r=1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\Longleftrightarrow m\\left( r\\right) =\\frac{1}\n1\\pm \\beta r^{2}};\\,\\beta \\geq 0.\n\\end{equation\nThen the corresponding two-dimensional PDM Lagrangian of (30) reads a\ntwo-dimensional Mathews-Lakshmanan type-I oscillato\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}\\frac{\\left[\n\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left( x_{1}^{2}+x_{2}^{2}\\right)\n\\right] }{1\\pm \\beta \\left( x_{1}^{2}+x_{2}^{2}\\right) }.\n\\end{equation}\n\nA Lagrangian of this type is neither separable nor superintegrable.\nNevertheless, our two-dimensional PDM Lagrangian $L\\left( x_{1},x_{2},\\dot{x\n_{1},\\dot{x}_{2};t\\right) $ (36) nonlocaly transforms into a \\emph\nsuperintegrable} two-dimensional Lagrangian $L\\left( q_{1},q_{2},\\tilde{q\n_{1},\\tilde{q}_{2};\\tau \\right) $ (22). Hence the two-dimensional PDM\\\nLagrangian $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ of (36)\nis a \\emph{sub-superintegrable \\ }PDM Lagrangian and the corresponding\nEuler-Lagrange equations (16) admit exact solution\n\\begin{equation}\nx_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) ;\\,\\Omega\n^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left( A_{1}^{2}+A_{2}^{2}\\right) }.\n\\end{equation\nLikewise, the reversed process is equally valid. That is, the relation \n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\emph{\\ Superintegrable}\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof \\ (22) with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega \\tau +\\varphi \\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=x_{j}\\sqrt{m\\left( r\\right) }\\,;\\smallskip \\medskip \\,j=1,2,$ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) }\\medskip $ \\\\ \n$f\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\medskip $ \\\\ \n$m\\left( r\\right) =1\/\\left( 1\\pm \\beta r^{2}\\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\n\\emph{Sub-superintegrable}\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (36) with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,\\medskip $\n\\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} .\n\\end{equation\nprovides the exact mapping from the \\emph{superintegrability} and exact\nsolvability of (22) into the \\emph{sub-superintegrability} and exact\nsolvability of the PDM Mathews-Lakshmanan type-I Lagrangian (36).\n\n\\subsubsection{Sub-superintegrable Mathews-Lakshmanan type-II PDM-oscillator\n}\n\nConsider a PDM particle $m\\left( r\\right) $ moving in a constant potential\nforce field of the for\n\\begin{equation}\nV\\left( x_{1},x_{2}\\right) =\\frac{1}{2}m\\left( r\\right) \\omega ^{2}\\left(\n\\xi _{1}^{2}+\\xi _{2}^{2}\\right)\n\\end{equation\nwith the corresponding Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left(\nr\\right) \\left[ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left( \\xi\n_{1}^{2}+\\xi _{2}^{2}\\right) \\right] .\n\\end{equation\nwhere $\\xi _{1},$ $\\xi _{2}\\in \n\\mathbb{R}\n$ are constants. This Lagrangian has the total energy $E_{tot}$ of (17) as\nthe only integral of motion and the corresponding Euler-Lagrange equations\nare non-separable. However, the substitution of $q_{j}=\\xi _{j}\\sqrt{m\\left(\nr\\right) }$ would nonlocaly transform it into the \\emph{superintegrable}\nLagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) $\nof (22) for $\\omega _{1}=\\omega _{2}=\\omega $. Under such settings\n\\begin{equation}\nq_{j}=\\int \\sqrt{m\\left( r\\right) }f\\left( r\\right) dx_{j}=\\xi _{j}\\sqrt\nm\\left( r\\right) }\\Longrightarrow f\\left( r\\right) =\\frac{\\xi _{j}}{2}\\frac\nm^{^{\\prime }}\\left( r\\right) }{m\\left( r\\right) }\\left( \\frac{x_{j}}{r\n\\right) ,\n\\end{equation\nand for $\\xi _{1}=\\xi _{2}=\\xi \/\\sqrt{2}$, this would imply tha\n\\begin{equation}\n2f\\left( r\\right) ^{2}=\\left( \\frac{m^{^{\\prime }}\\left( r\\right) }{2m\\left(\nr\\right) }\\right) ^{2}\\left( \\frac{\\xi _{1}^{2}x_{1}^{2}+\\xi\n_{2}^{2}x_{2}^{2}}{r^{2}}\\right) \\Longrightarrow f\\left( r\\right) =\\frac{\\xi\nm^{^{\\prime }}\\left( r\\right) }{4m\\left( r\\right) }.\n\\end{equation\nConsequently, the corresponding two-dimensional PDM Euler-Lagrange equation\n(17), fo\n\\begin{equation*}\nm\\left( r\\right) =\\frac{1}{1\\pm \\beta r^{2}},\n\\end{equation*\nread\n\\begin{equation}\n\\frac{d}{dt}\\left[ \\frac{\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}+\\omega ^{2}\\xi ^{2\n}{2(1\\pm \\beta r^{2})}\\right] =0,\n\\end{equation\nand admits solutions of the form\n\\begin{equation}\nx_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,;\\,\\Omega\n^{2}=\\frac{\\mp \\omega ^{2}\\beta \\xi ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\,;\\,\\beta \\geq 0;\\,j=1,2\\medskip \\medskip\n\\end{equation\nUnder such settings, one can show that for $\\beta =\\mp 1\/\\xi ^{2}$ the\nLagrangian of (36) and the Lagrangian of (40) indulge the very same\ndynamical properties as documented in the corresponding Euler-Lagrange\nequation (16). Hence, the Lagrangian at hand here is a Mathews-Lakshmanan\ntype-II PDM-oscillators Lagrangian.\n\nObviously, moreover, our non-separable and non-superintegrable Lagrangian\n(40) non-locally transforms into a separable and \\emph{superintegrable}\nLagrangian (22). Our PDM Lagrangian (40) is a \\emph{sub-superintegrable}\nMathews-Lakshmanan type-II PDM-oscillators Lagrangian, therefore. In short,\nthe relatio\n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\ \\emph{Superintegrable}\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof \\ (22) with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega \\tau +\\varphi \\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=\\xi _{j}\\sqrt{m\\left( r\\right) }\\medskip \\,;\\,j=1,2$ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) }\\medskip $ \\\\ \n$f\\left( r\\right) =\\frac{\\xi m^{^{\\prime }}\\left( r\\right) }{4m\\left(\nr\\right) }\\medskip ;\\,\\xi _{j}=\\frac{\\xi }{\\sqrt{2}}$ \\\\ \n$m\\left( r\\right) =1\/\\left( 1\\pm \\beta r^{2}\\right) ;\\beta =\\mp 1\/\\xi\n^{2}\\medskip \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\n\\emph{Sub-superintegrable}\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (39) with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,\\medskip\n\\medskip $ \\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} .\n\\end{equation\nrepresents the sought after mapping from\/to \\emph{superintegrable}\nLagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) $\nof \\ (22)\\ to\/from the \\emph{Sub-superintegrable}\\medskip\\ Lagrangian \nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $ of \\ (39).\n\n\\subsubsection{Sub-superintegrable Mathews-Lakshmanan type-III PDM\nshifted-oscillators}\n\nLet us now use a position-dependent mass with a different functional\nstructure moving in a shifted-oscillator force field and described by the\nLagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}m\\left(\nr_{s}\\right) \\left\\{ \\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}\\left[\n\\left( x_{1}+\\gamma _{1}\\right) ^{2}+\\left( x_{2}+\\gamma _{2}\\right) ^{2\n\\right] \\right\\} .\n\\end{equation\nwhere $r_{s}=\\sqrt{\\left( x_{1}+\\gamma _{1}\\right) ^{2}+\\left( x_{2}+\\gamma\n_{2}\\right) ^{2}}$ is introduced for convenience. The form of the shifted\noscillator potential is clear here. At this point, we may recollect that the\nfunctional nature\/structure of the position-dependent mass $m\\left( \\bar{x\n\\right) $ determines the nature\/structure of $g\\left( \\bar{x}\\right) $ and \n\\,f\\left( \\bar{x}\\right) $ in our nonlocal point transformation as suggested\nby equation (19). Therefore, $f\\left( r\\right) \\longrightarrow f\\left(\nr_{s}\\right) $ and $g\\left( r\\right) \\longrightarrow g\\left( r_{s}\\right) $\nfor our Lagrangian (46) at hand. Moreover, it is a straightforward manner,\nand in parallel with (32)-(35), one may show that the substitution o\n\\begin{equation}\nq_{j}=\\left( x_{j}+\\xi _{j}\\right) \\sqrt{m\\left( r_{s}\\right) \n\\Longrightarrow f\\left( r_{s}\\right) =1+\\frac{1}{4}\\frac{m^{^{\\prime\n}}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s}\n\\end{equation\nwould consequently, for the choice \n\\begin{equation}\nf\\left( r_{s}\\right) =m\\left( r_{s}\\right) -\\frac{1}{4}\\frac{m^{^{\\prime\n}}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s}=1+\\frac{1}{4}\\frac\nm^{^{\\prime }}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s},\n\\end{equation\nyiel\n\\begin{equation}\nm\\left( r_{s}\\right) =\\frac{1}{1\\pm \\beta r_{s}^{2}}=\\frac{1}{1\\pm \\beta\n\\left[ \\left( x_{1}+\\gamma _{1}\\right) ^{2}+\\left( x_{2}+\\gamma _{2}\\right)\n^{2}\\right] },\n\\end{equation\nThen the corresponding two-dimensional PDM Lagrangian of (46) reads a\ntwo-dimensional Mathews-Lakshmanan type-III PDM shifted-oscillators\nLagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}\\frac{\\left[\n\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\omega ^{2}r_{s}^{2}\\right] }{1\\pm \\beta\nr_{s}^{2}}.\n\\end{equation\nOur two-dimensional PDM Lagrangian $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $ (50) nonlocaly transforms into a \\emph{superintegrable}\ntwo-dimensional Lagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ of (22). Hence our PDM-Lagrangian $L\\left( x_{1},x_{2}\n\\dot{x}_{1},\\dot{x}_{2};t\\right) $ (50) is a \\emph{sub-superintegrable \nLagrangian and the corresponding Euler-Lagrange equation (16) admits exact\nsolutions of the for\n\\begin{equation}\nx_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) -\\gamma\n_{j};\\,\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) };\\,j=1,2.\n\\end{equation\nThe process is summed up a\n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\ Superintegrable\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof \\ (22) with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\omega \\tau +\\varphi \\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=x_{j}\\sqrt{m\\left( r_{s}\\right) }\\medskip \\,;\\,j=1,2$ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r_{s}\\right) }\\medskip $ \\\\ \n$f\\left( r_{s}\\right) =m\\left( r_{s}\\right) -\\frac{1}{4}\\frac{m^{^{\\prime\n}}\\left( r_{s}\\right) }{m\\left( r_{s}\\right) }r_{s}\\medskip \\medskip $ \\\\ \n$m\\left( r_{s}\\right) =1\/\\left( 1\\pm \\beta r_{s}^{2}\\right) \\medskip \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\nSub-superintegrable\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (50) with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) -\\gamma\n_{j}\\,\\medskip $ \\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} ,\n\\end{equation\nto represent the mapping from the \\emph{superintegrability} harmonic\noscillators (22) into \\emph{sub-superintegrability} of the above\nMathews-Lakshmanan type-III PDM shifted-oscillators (50).\n\n\\subsection{Superintegrable shifted-linear oscillators into\nsub-superintegrable nonlinear PDM-oscillators}\n\nConsider a \\emph{unit mass }particle moving in the two-dimensional\nshifted-oscillators potential \n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left[ \\alpha _{1}^{2}\\left( q_{1}+\\eta\n_{1}\\right) ^{2}+\\alpha _{2}^{2}\\left( q_{2}+\\eta _{2}\\right) ^{2}\\right]\n\\Longrightarrow V_{j}\\left( q_{j}\\right) =\\frac{1}{2}\\alpha _{j}^{2}\\left(\nq_{j}+\\eta _{j}\\right) ^{2};\\,j=1,2,\n\\end{equation\nin the generalized coordinates, with the constant shifts $\\eta _{1},$ $\\eta\n_{2}\\in \n\\mathbb{R}\n$. Then, the corresponding two-dimensional Lagrangia\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -\\frac{1}{2}\\left[ \\alpha\n_{1}^{2}\\left( q_{1}+\\eta _{1}\\right) ^{2}+\\alpha _{2}^{2}\\left( q_{2}+\\eta\n_{2}\\right) ^{2}\\right] ,\n\\end{equation\nyields the Euler-Lagrange equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\tilde{q}_{j}+\\alpha _{j}^{2}\\left( q_{j}+\\eta _{j}\\right)\n=0\\Longrightarrow q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\alpha _{j}\\tau\n+\\varphi \\right) -\\eta _{j};\\,j=1,2,\n\\end{equation\nwith the initial conditions that $\\,q_{j}\\left( 0\\right) =A_{j}-\\eta _{j},\\\n\\tilde{q}_{j}\\left( 0\\right) =0$, say. The separability of this Lagrangian\nis obvious. However, the verification of the superintegrability of such a\nLagrangian follows (step-by-step) from the complex factorization recipe\n(c.f., e.g., \\cite{42,43,44}) by introducing the complex function\n\\begin{equation}\nQ_{j}=\\tilde{q}_{j}+i\\alpha _{j}\\left( q_{j}+\\eta _{j}\\right)\n\\Longrightarrow \\frac{d}{d\\tau }Q_{j}=i\\alpha _{j}Q_{j};\\,\\,j=1,2.\n\\end{equation\nWhich would, in turn, suggest that \n\\begin{equation}\nQ_{jk}=Q_{j}^{\\alpha _{k}}\\left( Q_{k}^{\\ast }\\right) ^{\\alpha\n_{j}}\\Longrightarrow \\frac{d}{d\\tau }Q_{jk}=i\\left( \\alpha _{j}\\alpha\n_{k}-\\alpha _{k}\\alpha _{j}\\right) Q_{jk}=0.;\\,\\,j,k=1,2.\n\\end{equation\nThat is, the complex function $Q_{jk}$ represent \\emph{complex }constants of\nmotion with vanishing deformed\/rescaled-time evolution $\\tau $, Yet, it is\nan easy task to show that the two fundamental integrals $I_{1}$ and $I_{2}$\nare given through the relation \n\\begin{equation}\nQ_{jj}=\\tilde{q}_{j}^{2}+\\alpha _{j}^{2}\\left( q_{j}+\\eta _{j}\\right)\n^{2}\\Longrightarrow 2E_{1}=I_{1}=Q_{11},\\,2E_{2}=I_{2}=Q_{22},\n\\end{equation\nWhereas, for $\\alpha _{1}=\\alpha _{2}=\\alpha _{\\circ }$, for example\n\\begin{equation}\nQ_{12}=\\left[ \\tilde{q}_{1}\\tilde{q}_{2}+\\alpha _{\\circ }^{2}\\left(\nq_{1}+\\eta _{1}\\right) \\left( q_{2}+\\eta _{2}\\right) \\right] +i\\alpha\n_{\\circ }\\left[ \\left( q_{1}+\\eta _{1}\\right) \\tilde{q}_{2}-\\left(\nq_{2}+\\eta _{2}\\right) \\tilde{q}_{1}\\right] ,\n\\end{equation\nwhich identifies two real integrals of motion $I_{3}$ and $I_{4}$ such that \n\\begin{equation}\nI_{3}=\\func{Re}Q_{12}=\\tilde{q}_{1}\\tilde{q}_{2}+\\alpha _{\\circ }^{2}\\left(\nq_{1}+\\eta _{1}\\right) \\left( q_{2}+\\eta _{2}\\right) \\text{ \\ ; \\ \\ }I_{4}\n\\func{Im}Q_{12}=\\alpha _{\\circ }\\left[ \\left( q_{1}+\\eta _{1}\\right) \\tilde{\n}_{2}-\\left( q_{2}+\\eta _{2}\\right) \\tilde{q}_{1}\\right] .\n\\end{equation\nTherefore, our two-dimensional Lagrangian (54) admits \\emph\nsuperintegrability} in the generalized coordinates $\\left(\nq_{1},q_{2}\\right) $ and in the deformed\/rescaled time $\\tau $.\n\nNext, under the nonlocal transformation (20), along with the substitution\n\\begin{equation}\nq_{j}=x_{j}\\sqrt{m\\left( r\\right) }-\\eta _{j}\\Longrightarrow \\frac{dq_{j}}\ndx_{j}}=\\sqrt{m\\left( r\\right) }\\left[ 1+\\frac{m^{\\prime }\\left( r\\right) }\n2m\\left( r\\right) }\\left( \\frac{x_{j}^{2}}{r}\\right) \\right] \\Longrightarrow\nf\\left( r\\right) =1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left( r\\right) }{m\\left(\nr\\right) }r.\n\\end{equation\nThis result looks very much the same as that of $f\\left( r\\right) $ used in\n(33). Thus it would, again, with the assumption tha\n\\begin{equation*}\nf\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r=1+\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\Longrightarrow m\\left( r\\right) =\\frac{1}\n1\\pm \\beta r^{2}},\n\\end{equation*\nlead to the \\emph{sub-superintegrable }Mathews-Lakshmanan type-I\nPDM-oscillator (36). The corresponding Euler-Lagrange equation (16) admits\nexact solutions as those in (37). Then, the \\emph{sub-superintegrability} of\nour Mathews-Lakshmanan type-I PDM-Lagrangian (36) turns out to be a\nconsequence of the \\emph{superintegrability} of the linear oscillators (22)\nand\/or the \\emph{superintegrability} of the shifted-oscillators (54).\nLikewise, the \\emph{sub-superintegrable} and exact solvable\nMathews-Lakshmanan type-I PDM-Lagrangian (36) may very well be nonlocally\ntransformed into two \\emph{superintegrable} Lagrangians, a \\emph\nsuperintegrable} linear oscillator (22) and a \\emph{superintegrable}\nshifted-oscillator (54). That is, the relation \n\\begin{equation}\n\\left\\{ \n\\begin{tabular}{c}\n\\emph{\\ Superintegrable}\\medskip \\\\ \n$L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) \\medskip $ \\\\ \nof (54), with\\medskip \\\\ \n$q_{j}\\left( \\tau \\right) =A_{j}\\cos \\left( \\alpha _{j}\\tau +\\varphi \\right)\n-\\eta _{j}\n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{l}\n$\\text{Nonlocal transformation\\medskip }$ \\\\ \n$q_{j}=x_{j}\\sqrt{m\\left( r\\right) }-\\eta _{j}\\,;\\smallskip \\medskip\n\\,j=1,2, $ \\\\ \n$\\tilde{q}_{j}=\\dot{x}_{j}\\sqrt{m\\left( r\\right) }\\medskip $ \\\\ \n$f\\left( r\\right) =m\\left( r\\right) -\\frac{1}{4}\\frac{m^{^{\\prime }}\\left(\nr\\right) }{m\\left( r\\right) }r\\medskip $ \\\\ \n$m\\left( r\\right) =1\/\\left( 1\\pm \\beta r^{2}\\right) \n\\end{tabular\n\\right\\} \\Longleftrightarrow \\left\\{ \n\\begin{tabular}{c}\n\\emph{Sub-superintegrable}\\medskip \\\\ \n$L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) \\medskip $ \\\\ \nof \\ (36), with\\medskip \\\\ \n$x_{j}\\left( t\\right) =A_{j}\\cos \\left( \\Omega t+\\varphi \\right) \\,\\medskip $\n\\\\ \n$\\text{ \\ }\\Omega ^{2}=\\frac{\\omega ^{2}}{1\\pm \\beta \\left(\nA_{1}^{2}+A_{2}^{2}\\right) }\\medskip \n\\end{tabular\n\\right\\} .\n\\end{equation\nwould describe the mapping from the \\emph{superintegrable} Lagrangians (54)\ninto the \\emph{sub-superintegrable} ones of (36).\n\n\\subsection{Superintegrable Isotonic Oscillator into sub-superintegrable\nPDM-deformed Isotonic oscillator}\n\nA \\emph{\"unit mass\"} particle moving in a two-dimensional isotonic\noscillator potential fiel\n\\begin{equation}\nV(q_{1},q_{2})=\\frac{1}{2}\\left( \\omega _{1}^{2}q_{1}^{2}+\\omega\n_{2}^{2}q_{2}^{2}+\\frac{\\beta _{1}}{q_{1}^{2}}+\\frac{\\beta _{2}}{q_{2}^{2}\n\\right) \\Longrightarrow V_{j}\\left( q_{j}\\right) =\\frac{1}{2}\\left( \\omega\n_{j}^{2}q_{j}^{2}+\\frac{\\beta _{j}}{q_{j}^{2}}\\right) ,\n\\end{equation\nwhere $\\,\\omega _{j}=n_{j}\\omega _{\\circ };\\,\\,j=1,2$, is described by the\nLagrangia\n\\begin{equation}\nL\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q}_{2};\\tau \\right) =\\frac{1}{2\n\\left( \\tilde{q}_{1}^{2}+\\tilde{q}_{2}^{2}\\right) -\\frac{1}{2}\\left( \\omega\n_{1}^{2}q_{1}^{2}+\\omega _{2}^{2}q_{2}^{2}+\\frac{\\beta _{1}}{q_{1}^{2}}\n\\frac{\\beta _{2}}{q_{2}^{2}}\\right) ,\n\\end{equation\nwhich is known to be the \\emph{superintegrable} Smorodisky-Winternitz type\nLagrangian (c.f., e.g., \\cite{29,38,40,42,47}). Its \\emph{superintegrability}\ncan be verified through the two complex substitution\n\\begin{equation}\nQ_{j}=\\left( \\tilde{q}_{i}^{2}-\\omega _{j}^{2}q_{j}^{2}+\\frac{\\beta _{j}}\nq_{j}^{2}}\\right) +2i\\omega _{j}q_{j}\\tilde{q}_{j}\\Longrightarrow \\frac{d}\nd\\tau }Q_{j}=2i\\omega _{j}Q_{j},\\smallskip \\medskip \\,\\,j=1,2,\n\\end{equation\nthat satisfy (25) with $Q_{jk}$ representing complex constants of motion and\nleads to more than two integrals of motion that manifest \\emph\nsuperintegrability}. \\ Moreover, the corresponding Euler-Lagrange equations\nof which read two Ermakov-Pinney's like equation\n\\begin{equation}\n\\frac{d}{d\\tau }\\tilde{q}_{j}=-\\omega _{j}^{2}q_{j}+\\frac{\\beta _{j}}\nq_{j}^{3}},\n\\end{equation\nwith the corresponding exact solution\n\\begin{equation}\nq_{j}=\\sqrt{\\frac{A_{j}}{\\omega _{j}}\\sin \\left( \\omega _{j}\\tau +\\delta\n_{j}\\right) }\\Longrightarrow \\beta _{j}=-A_{j}^{2};\\,j=1,2.\n\\end{equation}\n\nYet, under the nonlocal transformation setting in (38) our \\emph\nsuperintegrable} Lagrangian $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ in (64) nonlocaly transforms into a \\emph\nsub-superintegrable} Smorodisky-Winternitz like PDM-oscillators Lagrangia\n\\begin{equation}\nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) =\\frac{1}{2}\\left\\{ \n\\frac{\\dot{x}_{1}^{2}+\\dot{x}_{2}^{2}-\\left( \\omega _{1}^{2}x_{1}^{2}+\\omega\n_{2}^{2}x_{2}^{2}\\right) }{1\\pm \\lambda \\left( x_{1}^{2}+x_{2}^{2}\\right) }\n\\left[ 1\\pm \\lambda \\left( x_{1}^{2}+x_{2}^{2}\\right) \\right] \\left( \\frac\n\\beta _{1}}{x_{1}^{2}}+\\frac{\\beta _{2}}{x_{2}^{2}}\\right) \\right\\} .\n\\end{equation\nThe Euler-Lagrange equation (16) for which admits exact solutions of the for\n\\begin{equation}\nx_{j}=\\sqrt{\\frac{A_{j}}{\\Omega }\\sin \\left( \\Omega \\tau +\\delta _{j}\\right) \n};\\,j=1,2\\medskip \\medskip ,\n\\end{equation\nwher\n\\begin{equation}\n\\Omega ^{2}=\\left\\{ \n\\begin{tabular}{ll}\n$\\omega ^{2}+\\lambda ^{2}\\left( A_{1}+A_{2}\\right) ^{2}$ & $;$ \\medskip\\ \n\\omega _{1}=\\omega _{2}=\\omega ,\\beta _{1}\\neq \\beta _{2}$ \\\\ \n$\\frac{A_{1}\\omega _{1}^{2}+A_{2}\\omega _{2}^{2}+\\lambda ^{2}\\left(\nA_{1}+A_{2}\\right) ^{3}}{A_{1}+A_{2}}\\medskip $ & $;$ $\\ \\omega _{1}\\neq\n\\omega _{2},\\beta _{1}\\neq \\beta _{2}$ \\\\ \n$\\left( \\omega _{1}^{2}+\\omega _{2}^{2}\\right) \/\\left( 16A^{2}\\lambda\n^{2}\\right) \\medskip $ & $;$ \\ $\\omega _{1}\\neq \\omega _{2},\\beta _{1}=\\beta\n_{2}\\Leftrightarrow A_{1}=A_{2}=A\n\\end{tabular\n\\right. .\n\\end{equation}\n\n\\section{Concluding Remarks}\n\nIn this article, and in parallel with our recent methodical proposal in \\cit\n{38}, we have introduced the two-dimensional extension of the\none-dimensional PDM-Lagrangians and their nonlocal transformation mappings'\nrecipes into constant \\emph{unit-mass} exactly solvable Lagrangians. Hereby,\nthe two-dimensional nonlocal point transformations (10) and the related\nEuler-Lagrange equations invariance conditions (19) are reported. However,\ndealing with Lagrangians in more than one-dimension renders \\emph\nsuperintegrability} to be unavoidably in the process. We have, therefore,\nasserted that, if a set of \\emph{\"superintegrable\nreference\/target-Lagrangians\"} $L\\left( q_{1},q_{2},\\tilde{q}_{1},\\tilde{q\n_{2};\\tau \\right) $ is mapped into a set of \\emph{\"PDM\ntarget\/reference-Lagrangians\"} $L\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x\n_{2};t\\right) $, then the set of \\emph{\"PDM target\/reference-Lagrangians\"} \nL\\left( x_{1},x_{2},\\dot{x}_{1},\\dot{x}_{2};t\\right) $\\ is a set of \\emph\n\"sub-superintegrable PDM-Lagrangians\"} (as shall be so labelled\nhereinafter). Two sets of illustrative examples are used. The first set is\ndevoted to the mappings from \\emph{superintegrable} linear oscillators into \n\\emph{sub-superintegrable} nonlinear PDM-oscillators. Where, three\ntwo-dimensional \\emph{sub-superintegrable} Mathews-Lakshmanan type\nPDM-oscillators are used: (i) a \\emph{sub-superintegrable}\nMathews-Lakshmanan type-I PDM-oscillator for a PDM-particle moving in a\nharmonic oscillator potential, (ii) a \\emph{sub-superintegrable}\nMathews-Lakshmanan type-II PDM-oscillator for a PDM-particle moving in a\nconstant potential, and (iii) a \\emph{sub-superintegrable} shifted\nMathews-Lakshmanan type-III PDM-oscillator for a PDM-particle moving in a\nshifted harmonic oscillator potential. The second set, nevertheless,\nincludes some \\emph{superintegrable} shifted linear oscillators (new to the\nbest of our knowledge) and isotonic oscillators that are mapped into \\emph\nsub-superintegrable} PDM-nonlinear and \\emph{sub-superintegrable}\nPDM-isotonic oscillators, respectively. The mappings included exact\nsolvability as well. Our observations are in order.\n\nWhilst the two-dimensional PDM Mathews-Lakshmanan type-I and type-III, and\nthe PDM shifted nonlinear oscillators share the same total energ\n\\begin{equation}\nE_{tot}=\\frac{1}{2}\\omega ^{2}\\frac{\\left( A_{1}^{2}+A_{2}^{2}\\right) }{1\\pm\n\\beta \\left( A_{1}^{2}+A_{2}^{2}\\right) },\n\\end{equation\nthe two-dimensional PDM Mathews-Lakshmanan type-II oscillators admit total\nenerg\n\\begin{equation}\nE_{tot}=\\frac{\\omega ^{2}\\xi ^{2}}{1-\\xi ^{2}\\left(\nA_{1}^{2}+A_{2}^{2}\\right) },\n\\end{equation\nand the two-dimensional PDM-deformed isotonic oscillators indulge total\nenerg\n\\begin{equation}\nE_{tot}=\\frac{1}{2}\\left\\{ \\frac{\\omega _{1}^{2}A_{1}+\\omega _{2}^{2}A_{2}}\n\\Omega \\pm \\lambda \\left( A_{1}+A_{2}\\right) }-\\left[ \\Omega \\pm \\lambda\n\\left( A_{1}+A_{2}\\right) \\right] \\left( A_{1}+A_{2}\\right) \\right\\} .\n\\end{equation\nYet, the two-dimensional PDM Mathews-Lakshmanan type-I and type-III inherit\nthe dynamical properties and trajectories of each other. On the other hand,\nthe \\emph{sub-superintegrability} of the PDM Mathews-Lakshmanan type-I\noscillators (36) may very well be attributed to the \\emph{superintegrability}\nof the linear oscillator (22) and\/or the shifted-oscillators (54).\n\nFinally, the generalization of the current methodical proposal into more\nthan two-dimensional recipes looks eminent and feasible. This is very\nobviously documented in the description of our equations (6) to (12), where \nj=1,2$ is used. Strictly speaking, for a three-dimensional case \nj=1,2\\rightarrow $ $j=1,2,3$ and $\\bar{x}=x_{1},x_{2}\\rightarrow \\bar{x\n=x_{1},x_{2},x_{3}$ in (10) and so on so forth. It would be interesting to\nstudy and explore the consequences of such generalization.\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:auto-hsp}\nThe main result of this paper is the Theorem ~\\ref{theo:cycle-graph-auto-weak-fails} and the Corollary ~\\ref{cor:cycle-graph-auto-strong-fails} where it is shown that both the weak and strong quantum Fourier samplings are guaranteed to always fail for the classically trivial cycle graph automorphism problem. It is also shown how to systematically determine the non-trivial classes of graphs for which quantum Fourier transform (QFT) always fails to construct the automorphism groups. Here, the term `non-trivial' refers to the classes of graph automorphism problems which can be solved trivially on a classical computer. This result puts an end to the decades long effort of finding a hidden subgroup algorithm for the graph isomorphism problem. Previously, the researchers have been proving increasing negative results which indicated that the probability of successfully deciding a graph isomorphism problem is exponentially small. This paper gives an algorithm to create arbitrarily large classes of graph automorphism problems for which the quantum hidden subgroup approach will always fail to compute the automorphism group no matter how big the computer or how small the size of the problem is. As the graph automorphism problem is Karp-reducible \\cite{kobler2012graph} to the graph isomorphism problem, it can be inferred that there are classes of graph isomorphism problem for which hidden subgroup approach will always fail irrespective of the size of the problem. The linear representation theory of the dihedral groups plays a very important role in proving this result.\n\nThe general framework for the hidden subgroup problems was first formulated in \\cite{brassard1997exact, hoyer2000quantum, mosca1999hidden}. The hidden subgroup problem can be defined as follows \\footnote{This pedagogically convenient version of HSP has been borrowed from the presentation titled 'Graph isomorphism, the hidden subgroup problem and identifying quantum states' by Pranab Sen.}.\n\n\\begin{mydef}[Hidden subgroup problem]\n\n{\\bf Given:} $G:$ group, $S:$ set, $f : G \\to S$ via an oracle.\\\\\n{\\bf Promise:} Subgroup $H \\le G$ such that $f$ is constant on the left\ncosets of $H$ and distinct on different cosets.\\\\\n{\\bf Task:} Find the hidden subgroup $H$ by querying $f$.\n\\end{mydef}\n\nThe hidden subgroup version of the graph isomorphism problem was first defined in \\cite{jozsa1998quantum}.The $n$-vertex graph isomorphism problem for rigid graphs of  $n$ vertices can be expressed as a hidden subgroup problem over the ambient symmetric group $S_{2n}$ or more specifically the wreath product $S_n \\wr \\mathbb{Z}_2$ where the hidden subgroup is promised to be either trivial or of order two \\cite{moore2010impossibility}.  The scheme and notation of the following definitions of the hidden subgroup problem, used in this paper, are borrowed from \\cite{lomonacopers2002, van2012quantum}. Erd{\\H{o}}s et al \\cite{erdHos1963asymmetric} have shown that the automorphism groups of the most of the graphs are trivial.  So, although, the problem was defined for all simple undirected graphs in \\cite{jozsa1998quantum}, this paper follows the example of \\cite{grigni2001quantum} and limit the discussion to the rigid graphs with trivial automorphism groups. The rigidity of the graphs in this definition will be temporarily relaxed in Section ~\\ref{sec:iso-auto} to prove reducibility.\n\n\\begin{mydef} [Graph isomorphism as a hidden subgroup problem ($\\text{{\\bf GI}}_{\\text{HSP}}$)]\nLet the $2 n$ vertex graph $\\Gamma = \\Gamma_1 \\sqcup \\Gamma_2$ be the disjoint union of the two rigid graphs $\\Gamma_1$ and $\\Gamma_2$ such that $Aut \\left(\\Gamma_1\\right) = Aut \\left(\\Gamma_2\\right) = \\left\\{e\\right\\}$. A map $\\varphi : S_{2n} \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$ \\footnote{$\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is the algebra of all $N \\times N$ matrices over the complex numbers $\\mathbb{C}$.} from the group $S_{2n}$ is said to have hidden subgroup structure if there exists a subgroup $H_\\varphi$ of $S_{2n}$, called a hidden subgroup, an injection $\\ell_\\varphi : S_{2n}\/H \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$, called a hidden injection, such that the diagram\n\\[\n\\begin{tikzcd}\nS_{2n} \\arrow{r}{\\varphi} \\arrow{d}{\\nu} & \\text{Mat}\\left(\\mathbb{C}, N \\right)\\\\\nS_{2n}\/H \\arrow{ur}{\\ell_\\varphi}&\n\\end{tikzcd}\n\\]\nis a commutative diagram, where $S_{2n}\/H_{\\varphi}$ denotes the collection of right cosets of $H_\\varphi$ in $S_{2n}$, and where $\\nu : S_{2n}\/H_\\varphi$ is the natural map of $S_{2n}$ onto $S_{2n}\/H_\\varphi$. The group $S_{2n}$ is called the ambient group and  the set $\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is called the target set.\n\nThe hidden subgroup version of the graph isomorphism problem is to determine a hidden subgroup $H$ of $S_{2n}$ with the promise that $H$ is either trivial or $|H| = 2$.\n\\end{mydef}\n\nThis section also gives a formal definition for the hidden subgroup representation of the graph automorphism problem.\n\n\\begin{mydef} [Graph automorphism as a hidden subgroup problem ($\\text{{\\bf GA}}_{\\text{HSP}}$)]\nFor a graph $\\Gamma$ with $n$ vertices, a map $\\varphi : S_{n} \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$ \\footnote{$\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is the algebra of all $N \\times N$ matrices over the complex numbers $\\mathbb{C}$.} from the group $S_{n}$ is said to have hidden subgroup structure if there exists a subgroup $\\text{Aut}\\left(\\Gamma\\right)$ of $S_{n}$, called a hidden subgroup, an injection $\\ell_\\varphi : S_{n}\/\\text{Aut}\\left(\\Gamma\\right) \\to \\text{Mat}\\left(\\mathbb{C}, N \\right)$, called a hidden injection, such that for each $g \\in \\text{Aut}\\left(\\Gamma\\right)$, $g \\left(\\Gamma\\right) = \\Gamma$ and, the diagram\n\\[\n\\begin{tikzcd}\nS_{n} \\arrow{r}{\\varphi} \\arrow{d}{\\nu} & \\text{Mat}\\left(\\mathbb{C}, N \\right)\\\\\nS_{n}\/\\text{Aut}\\left(\\Gamma\\right) \\arrow{ur}{\\ell_\\varphi}&\n\\end{tikzcd}\n\\]\nis commutative, where $S_{n}\/\\text{Aut}\\left(\\Gamma\\right)$ denotes the collection of right cosets of $\\text{Aut}\\left(\\Gamma\\right)$ in $S_{n}$, and where $\\nu : S_{n}\/\\text{Aut}\\left(\\Gamma\\right)$ is the natural map of $S_{n}$ onto $S_{n}\/\\text{Aut}\\left(\\Gamma\\right)$.  $S_{n}$ is called  the ambient group and  $\\text{Mat}\\left(\\mathbb{C}, N \\right)$ is called  the target set.\n\nThe hidden subgroup version of the graph automorphism problem is to determine a hidden subgroup $\\text{Aut}\\left(\\Gamma\\right)$ of $S_{n}$ with the promise that $\\text{Aut}\\left(\\Gamma\\right)$ is either of trivial or non-trivial order depending on the type of $\\Gamma$.\n\\end{mydef}\n\n\\subsection{Outline of the paper}\nFor the convenience of the readers, a brief outline of the paper is given here. The Section ~\\ref{sec:Historical-context} gives an overview of the related literature, Section  ~\\ref{sec:prem} provides the preliminary background needed to follow the discussion used in this paper, Section ~\\ref{sec:qft-gi} provides the already known results on the hidden subgroup approach for graph isomorphism, Section ~\\ref{sec:weak-cycle} presents the original result that the hidden subgroup approach is guaranteed to fail for an easy class of graph automorphism problem, and Section ~\\ref{sec:qft-ga-plus} presents another original result which is a systematic way to build arbitrarily  large classes of graph automorphism problems for which hidden subgroup algorithms are guaranteed to fail. Finally, it has been discussed whether the hidden subgroup algorithms are the most appropriate ways to attempt combinatorial problems in quantum computation.\n\n\\subsection{Key technical ideas}\nThis section summarizes the key technical ideas used to prove the main results of this paper.  The automorphism group of a cycle graph is the dihedral group $D_n$ of order $2 n$. It has been shown in  Lemma ~\\ref{lem:d-n-1-d-rep-prob-zero} that the probability of measuring the labels of one dimensional irreducible representations of $D_n$ is zero for non-trivial representations. Then, in Lemma ~\\ref{lem:d-n-2-d-rep-prob-zero},  it has been shown that the probability of measuring the labels of two dimensional irreducible representations of $D_n$ is always zero. Combining these two lemmas, it has been proved in Theorem ~\\ref{theo:cycle-graph-auto-weak-fails} that Weak quantum Fourier sampling always fails to solve the cycle graph automorphism problem irrespective of its size. Finally, this paper gives Algorithm ~\\ref{algo:arb-fail} to create arbitrarily large class of graph automorphism problems (with rotational symmetries) for which quantum Fourier transform is guaranteed to always fail irrespective of the size of the graphs.\n\nIt was already proved in \\cite{grigni2001quantum} that single coset state can provide only exponentially less information for the graph isomorphism problem. Hence, later works, for example  \\cite{hallgren2010limitations, moore2010impossibility}, investigated the possibility of using multiple copies of coset states. The current work gives a stronger result on the graph automorphism problem for the single coset state. Moreover, all the multi-coset state algorithms are conditioned on the successful execution of weak sampling, which has been proved in this paper to fail with guarantee for the problem of interest. So, the case of failure can also be inferred for multi-coset approaches, e.g., the sieve algorithms. To summarize, if there are rotational symmetries in the graph, we will not get exponentially less information with a single copy coset state rather, but, even worse, we will get exactly zero amount of information no matter how large the quantum computer is. Same would be true for k-copy coset states.\n\n\\section{Historical context}\n\\label{sec:Historical-context}\nRead et al. \\cite{Read1977} have named the tendency of incessant but unsuccessful attempts at the graph isomorphism problem as the {\\it graph isomorphism disease}. This indicates the amount of interest about the problem among the researchers. For almost three decades, until $2015$, the  best known algorithm for the general graph isomorphism problem has been due to Babai et al. \\cite{babai1983canonical}. The algorithm exploits graph canonization techniques through label reordering in exponential time (\\ensuremath{exp \\left(n^{\\frac{1}{2} + o\\left(1\\right)}\\right)}), where \\ensuremath{n = |V|}. Faster algorithms have been proposed for graph sub classes with special properties. In \\cite{babai1983canonical}, Babai et al. also proved the bound for tournament graphs is \\ensuremath{n^{\\left(\\frac{1}{2} + o\\left(1\\right)\\right) \\log n}}. In \\cite{luks1982isomorphism}, Luks reduced the bounded valence graph isomorphism problem to the color automorphism problem, and gave a polynomial time algorithm. In another paper \\cite{babai1982isomorphism}, Babai et al. created two polynomial algorithms using two different approaches, i.e., the tower of groups method, and the recursion through systems of imprimitivity respectively, for the bounded eigenvalue multiplicity graph isomorphism problem. The isomorphism problem for planar graphs is known to be in polynomial time due to Hopcroft et al. \\cite{hopcroft1974linear}. In their paper, the authors used a reduction approach to eventually tranform the graphs into five regular polyhedral graphs and check the isomorphism by exhaustive matching in a fixed finite time. Miller \\cite{miller1980isomorphism} used a different approach by finding minimal embeddings of the graphs of bounded genus and checking their isomorphism by generating codes. Babai et al. in \\cite{babai1980random} and Czajka et al. in \\cite{czajka2008improved} showed that the isomorphism of almost all the graphs in a class of random graphs can be tested in linear time. Both of their approaches exploit the properties of the degree sequence of a random graph. Babai et al. \\cite{babai2013faster} proved that while the graph isomorphism problem for strongly regularly graphs may be solved faster than the general version it is still an exponential time algorithm. A series of dramatic events took place recently between $2015$ and $2017$ in the field of graph isomorphism. In December, $2015$ \\cite{babai2015graph}, Babai posted a pre-print claiming that the general graph isomorphism problem can be solved in quasipolynomial time. One of the authors of this papers was fortunate enough to witness a live proof session of the algorithm by Babai in Discrete Mathematics $2016$. Two years later, Helfgott \\cite{helfgott2017isomorphismes} pointed out a serious error in that proof. Babai immediately fixed the proof and graph isomorphism still remains in quasipolynomial time.\n\nAlthough there is a quasi-polynomial time algorithm for the general graph isomorphism problem, it is not proven to be optimal. So, the complexity class of the graph isomorphism problem is yet undecided. While it is known that the problem is in {\\bf NP} \\cite{garey2002computers}, it is not known whether the problem is in {\\bf P} or {\\bf NP}-complete. This is why the graph isomorphism problem is called an {\\bf NP}-intermediate problem. Sch{\\\"o}ning \\cite{schoning1988graph} has shown that graph isomorphism is in \\ensuremath{L^P_2} and not \\ensuremath{\\gamma}-complete under the assumption that the polynomial hierarchy does not collapse to \\ensuremath{L^P_2}. Given this information, many researchers believe that the graph isomorphism problem is not {\\bf NP}-complete.\n\nWhile the efforts towards finding an efficient solution for the general graph isomorphism problems have been unsuccessful, the researchers have attempted practically feasible methods to solve the problem in reasonable time frame.\n\nThe hidden subgroup approach for both the graph isomorphism and automorphism problems require the computing of the quantum Fourier sampling of the ambient symmetric group. This has been an active area of research since Peter Shor invented the famous Shor's algorithm, a quantum hidden subgroup algorithm for the abelian groups,  to solve prime factorization \\cite{shor1999polynomial}. While at this moment, there is no known efficient quantum hidden subgroup algorithm for symmetric groups, researchers have shed some light on why it had been so difficult to find them.\n\nWhile surveys like \\cite{RevModPhys.82.1}, summarizes the advances made so far in the area of hidden subgroup algorithms, it would always be helpful to review the negative results in this section to illustrate why this is a difficult problem. It is noteworthy that all the positive results, so far, have been demonstrated for the synthetically created product groups. While this approach may not have immediate practical application, this idea of creating synthetic groups has been used in this paper to generalize results. One of the first results for the non-abelian hidden subgroup problems was presented by Roetteler et al \\cite{roetteler1998polynomial}. In that paper, the authors proved an efficient hidden subgroup algorithm for the wreath product $\\mathbb{Z}^k_2 \\wr \\mathbb{Z}_2$ which is a non-abelian group. Similarly, Ivanyos et al \\cite{ivanyos2003efficient} proved the existence of an efficient hidden subgroup algorithm for a more general non-abelian nil-$2$ groups. Later Friedl et al  \\cite{friedl2003hidden} generalized the result such that there are efficient hidden subgroup algorithms for the groups whose\nderived series have constant length and whose Abelian factor groups are each the\ndirect product of an Abelian group of bounded exponent and one of polynomial\nsize. Ettinger et al \\cite{ettinger2000quantum} showed that it is possible to reconstruct a subgroup hidden inside the dihedral group using finite number of queries. This result was later generalized by Ettinger et al \\cite{ettinger1999hidden} that arbitrary groups may be reconstructed using finite queries but they did not give any specific set of measurement.\n\nIn \\cite{moore2002hidden}, Moore et al proved that although weak quantum Fourier sampling fails to determine the hidden subgroups of the non-abelian groups of the form $\\mathbb{Z}_q \\ltimes \\mathbb{Z}_p$, where $q \\mid \\left(p-1\\right)$ and $q = p \/ \\text{polylog}\\left(p\\right)$, strong Fourier sampling is able to do that. Later on, Moore et al  \\cite{moore2006generic, moore2005explicit}  proved the existence of $polylog \\left(|G|\\right)$ sized quantum Fourier circuits for the groups like $S_n$, $H \\wr S_n$, where $|H| = \\text{poly}\\left(n\\right)$, and the Clifford groups. The authors also gave the circuits of subexponential size for standard groups like $\\text{GL}_n \\left(q\\right)$, $\\text{SL}_n \\left(q\\right)$, $\\text{PGL}_n \\left(q\\right)$, and $\\text{PSL}_n \\left(q\\right)$, where $q$ is a fixed prime power. Moore et al  \\cite{moore2008symmetric} have also presented a stronger result where they have shown that it is not possible to reconstruct a subgroup hidden inside the symmetric group with strong Fourier sampling and both arbitrary POVM and entangled measurement. At the same time, the authors did not rule out the possibility of success using other possible measurements which is still an open question. Bacon et al \\cite{bacon2005optimaldi} proved that the so called {\\it pretty good measurement} is optimal for the dihedral hidden subgroup problem. Moore et al  \\cite{moore2005distinguishing} extended this result for the case where the hidden subgroup is a uniformly random conjugate of a given subgroup. Moore et al \\cite{moore2007power} eventually proved a more general results that strong quantum Fourier sampling can reconstruct $q$-hedral groups. Alagic et al \\cite{alagic2005strong} proved a general result that strong Fourier sampling fails to distinguish the subgroup of the power of a given non-abelian simple group. Moore et al \\cite{moore2005tight} later proved that arbitrary entangled measurement on $\\Omega \\left(n \\log n\\right)$ coset states is necessary and sufficient to extract non-negligible information. Similar result was also proved in \\cite{hallgren2010limitations} separately. Few years later, Moore et al \\cite{moore2010impossibility} proved a negative result that the {\\it quantum sieve algorithm}, i.e. highly entangled measurements across $\\Omega(n \\log n)$ coset states, cannot solve the graph isomorphism problem.\n\nIt is important to point out that all the groups used in the previously mentioned results are conveniently chosen and synthetically created. Moreover, they are sporadic so it is not clear how the knowledge can be extrapolated to the symmetric groups. As the graph automorphism problem is Karp-reducible to the graph isomorphism problem, it is believed to be sufficient to investigate the hidden subgroup representation of the graph isomorphism problem. With all these unsuccessful attempts for the last couple of decades presented above, one may ask whether the hidden subgroup approach is the right way to attempt the graph isomorphism problem. If it is, there would have been a Karp-reduction from the hidden subgroup representation of the graph automorphism problem to the hidden subgroup representation of the graph isomorphism problem. This paper gives one such reduction in Section ~\\ref{sec:iso-auto}. So, another way of looking at the problem is to understand the hidden subgroup  complexity of the graph automorphism problem and reduce the results to graph isomorphism.\n\n\\section{Preliminaries}\n\\label{sec:prem}\n\\subsection{Graph Theory}\nMost of the work presented in this paper involves the graph isomorphism and automorphism problems. So, it would be appropriate to start the background section with a few concepts of graph theory.  The materials in this section are reproduced from the very well written book by Bollob\\'{a}s \\cite{bollobas2013modern}. The section does not contain a comprehensive coverage on graph theory, rather they are only related to the discussion of this paper.\n\n\\begin{mydef} [Graph]\nA graph $\\Gamma$ is an ordered pair of disjoint sets $\\left(V, E\\right)$ such that $E$ is a subset of the set $V^{\\left(2\\right)}$ of unordered pairs of $V$.\n\\end{mydef}\n\n$V$ is known as the set of vertices, and $E$ is known as the set of edges. Each element of $E$ connects two elements of $V$. A graph is directed if the edge $\\left(v_i, v_j\\right)$ is an element of $E$ but $\\left(v_j, v_i\\right)$ is not for all $i$ and $j$. A simple graph does not have loops or multi-edges. This paper only focuses on questions defined on simple undirected graphs.\n\n\\subsubsection{Graph isomorphism and automorphism}\n\\label{sec:iso-auto}\nThe graph isomorphism and automorphism problems are the two of the oldest problems in combinatorics. The formal statement of the graph isomorphism problem goes as follows as mentioned in \\cite{fortin1996graph}. \n\n\\begin{mydef} [Graph isomorphism ({\\bf GI})]\n\\label{prob:gi}\nGiven two graphs, \\ensuremath{\\Gamma_1 = \\left(V_1, E_1\\right)} and \\ensuremath{\\Gamma_2 = \\left(V_2, E_2\\right)},  does there exist a bijection \\ensuremath{f : V_1 \\to V_2} such that \\ensuremath{\\forall a, b \\in V_1, \\left(a, b\\right) \\in E_1 \\iff \\left(f\\left(a\\right), f\\left(b\\right)\\right) \\in E_2}?\n\\end{mydef}\nHere, \\ensuremath{V_1} and \\ensuremath{V_2} are the sets of vertices and \\ensuremath{E_1} and \\ensuremath{E_2} are the sets of edges of \\ensuremath{\\Gamma_1} and \\ensuremath{\\Gamma_2} respectively.\n\nThe graph automorphism problem is a special version of Definition \\ref{prob:gi} when $\\Gamma_1 = \\Gamma_2$.  \n\n\\begin{mydef} [Graph automorphism ({\\bf GA})]\n\\label{prob:ga}\nGiven a graph $\\Gamma = \\left(V, E\\right)$, compute the automorphism groups which are $\\Gamma \\to \\Gamma$ isomorphisms; and form the subgroup $\\text{Aut}\\left(\\Gamma\\right)$ of the symmetric group $S_{|V|}$.\n\\end{mydef}\n\nThe reducibility from {\\bf GA} to {\\bf GI} is discussed in the rest of this section based on a few theorems proven in \\cite{kobler2012graph}.  The outline is as follows. First, it has been shown that {\\bf GA} is Turing-reducible to {\\bf GI} i.e. $\\text{\\bf GA} \\le^p_T \\text{\\bf GI}$ using Algorithm ~\\ref{algo:ga-gi-turing-reduction} which is the Example 1.10 of \\cite{kobler2012graph}. \n\n\\begin{algorithm}[H]\n\\caption{$GA \\le^p_T GI$}\n     \\label{algo:ga-gi-turing-reduction}\n\\begin{algorithmic}[1]\n\\Procedure {GA-GI-Turing-Reduction}{$\\Gamma, n$} \\Comment{graph $\\Gamma$ with $n$ nodes}\n\n\\For{$i\\gets 1, n-1 $}\n\\For{$j\\gets i+1, n $}\n\\If{$\\left(\\Gamma_{[i]}, \\Gamma_{[j]}\\right) \\in GI$} \\Comment{$\\Gamma_{[i]}$ denotes a copy of the graph $\\Gamma$ with a label attached with node $i$}\n\\State {\\bf accept}\n\\EndIf\n\\EndFor\n\\EndFor\n\\State {\\bf reject};\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nThen, it has been proven that {\\bf GA} is Karp-reducible to {\\bf GI}. Instead of reproducing the detailed proof from \\cite{kobler2012graph}, this section provides a sketch of it. First, it needs to be shown that {\\bf GA} has a polynomial time computable {\\it or}-function. Then, it has to be shown that {\\bf GI} has both polynomial time computable {\\it and-} and {\\it or}-functions. Combining these results, it can be argued that {\\bf GA} is Karp-reducible to {\\bf GI} i.e. $\\text{\\bf GA} \\le^p_m \\text{\\bf GI}$.\n\nAt this point, it is natural to ask whether $\\text{\\bf GA}_{HSP} \\le^p_T \\text{\\bf GI}_{HSP}$ or ${\\bf GA}_{HSP} \\le^p_m {\\bf GI}_{HSP}$.\n\nIt can be trivially shown that the hidden subgroup representation of the graph automorphism problem is Turing-reducible to the graph isomorphism problem by giving two input graphs as the original graph and the candidate automorphism of the original graph. The technique to prove the Karp-reducibility from $\\text{\\bf GA}_{HSP}$ to $\\text{\\bf GI}_{HSP}$ was kindly shown to the authors in a public forum by Grochow \\cite{Grochow2017}. The sketch of the algorithm is given below.\n\nFirst, the condition on the the rigidity of the input graphs is relaxed. This makes the case harder.  Instead, the (non-rigid) {\\bf GI} as an HSP is described in the same way, but now the goal is to determine the size of the hidden subgroup, or a generating set. The difference between the isomorphic and non-isomorphic cases will be a factor of $2$ in the order of the hidden subgroup. If  the problem is expressed as finding generators of the hidden subgroup, then the question is whether any generator switches $\\Gamma_1$ and $\\Gamma_2$.\n\nNow, an instance of $\\text{\\bf GA}_{HSP}$ corresponding to a graph $\\Gamma$ is given by the function from $S_n \\to M_n(\\mathbb{C})$ defined by $f(\\pi) = A(\\pi(\\Gamma))$ where $A(\\cdot)$ denotes the adjacency matrix. In particular, $f(e) = A(\\Gamma)$. Then the usual Karp reduction is applied from {\\bf GA} to {\\bf GI} to get a pair of graphs $\\Gamma_1, \\Gamma_2$. Then an instance of $\\text{\\bf GI}_{HSP}$ instance, of the type described in the preceding paragraph, can be created corresponding to the pair $\\Gamma_1, \\Gamma_2$ (that is, the disjoint union $\\Gamma_1 \\cup \\Gamma_2$). Thus ${\\bf GA}_{HSP} \\le^p_m {\\bf GI}_{HSP}$.\n\n\\subsection{Representation Theory}\nA few concepts of the representation theory are discussed in the current section which are relevant to this paper. The discussion is limited to the representation theory of symmetric groups. A more detailed introduction may be found in \\cite{fulton1991representation, curtis1966representation, sagan2013symmetric}. The paper has borrowed the notations and definitions of representation theory generously from the above mentioned standard resources.\n\n\\begin{mydef} [Matrix representations \\cite{sagan2013symmetric}]\nA matrix representation of a group $G$ is a group homomorphism\n\\begin{align}\nX : G \\to GL_d.\n\\end{align}\nEquivalently, to each $g \\in G$ is assigned $X \\left(g\\right) \\in \\text{Mat}_d$ such that\n\\begin{itemize}\n\\item $X \\left(e\\right) = I$ the identity matrix, and\n\\item $X \\left(g h\\right) = X \\left(g\\right) X \\left(h\\right)$ for all $g, h \\in G$.\n\\end{itemize}\nThe parameter $d$ is called the degree, or dimension, of the representation and is denoted by $\\text{deg } X$.\n\\end{mydef}\n\n\\begin{mydef} [Young diagram]\nFor any partition $\\lambda_1, \\ldots, \\lambda_k$ of an integer $\\lambda$, there is a diagram associated called the Young diagram where there are $\\lambda_i$ cells in the $i$-th row. The cells are lined up on the left.\n\\end{mydef}\n\n\\begin{mydef} [Restricted and induced representations]\nIf $H \\subset G$ is a subgroup, any representation $\\rho_1$ of $G$ restricts to a representation of $H$, denoted $Res^G_H \\rho_1$ or simple $Res \\rho_1$. Let $\\rho_2 \\subset \\rho_1$ be a subspace which is $H$-invariant. For any $g$ in $G$, the subspace $g . \\rho_2 = \\left\\{g . w : w \\in \\rho_2\\right\\}$ depends only on the left coset of $g H$ of $g$ modulo $H$, since $g h . W = g . \\left(h . \\rho_2\\right) = g. \\rho_2$; for a coset $c$ in $G\/H$,  $c . \\rho_2$ is the subspace of $\\rho_1$ subspace of $\\rho_1$. $\\rho_1$ is induced by $\\rho_2$ if every element in $\\rho_1$ can be written uniquely as a sum of elements in such translates of $\\rho_2$, i.e. \n\\begin{align}\n\\rho_1 &= \\bigoplus_{c \\in G\/H} c . \\rho_2\n\\end{align}\nIn this case, the induced representation is $\\rho_1 = Ind^G_H \\rho_2$  = Ind $\\rho_2$.\n\\end{mydef}\n\nA common representation to be seen in later sections of this report is the regular representation \\cite{fulton1991representation}.\n\n\\begin{mydef} [Regular representation]\nIf $X$ is any finite set and $G$ acts on the left on $X$, i.e., $G \\to Aut \\left(X\\right)$ is a homomorphism to the permutation group of $X$, there is a associated permutation representation: let $V$ be the vector space with basis $\\left\\{e_x: x \\in X\\right\\}$, and let $G$ act on $V$ by\n\\begin{align}\ng \\cdot \\sum a_x e_x = \\sum a_x e_{g x}.\n\\end{align}\nThe regular representation, denoted $R_G$ or $R$, corresponds to the left action of $G$ on itself.\n\\end{mydef}\n\nThe character of a group element is defined as follows \\cite{fulton1991representation}.\n\n\\begin{mydef} [Character]\nIf $\\rho$ is a representation of a group $G$, its character $\\chi_\\rho$ is the complex-valued function on the group defined by \n\\begin{align}\n\\chi_\\rho \\left(g\\right) &= Tr \\left(g|_\\rho\\right),\n\\end{align}\nthe trace of $g$ on $\\rho$.\n\\end{mydef}\n\nIt is also useful to define the inner product of characters \\cite{sagan2013symmetric}.\n\n\\begin{mydef}[Inner product of characters]\n\\label{def:inner-prod-char}\nLet $\\chi$ and $\\psi$ be the characters of a group $G$. The {\\it inner product} of $\\chi$ and $\\Psi$ is\n\\begin{align}\n\\langle \\chi, \\Psi\\rangle &= \\frac{1}{|G|} \\sum_{g\\in G} \\chi \\left(g\\right) \\Psi^\\dagger \\left(g\\right)\n\\end{align}\n\\end{mydef}\n\n\n\nThe character table of a finite group  is defined as follows \\cite{sagan2013symmetric}.\n\n\\begin{mydef} [Character table]\nLet $G$ be a group. The {\\it character table} of $G$ is an array with rows indexed by the inequivalent irreducible characters of $G$  and columns indexed by the conjugacy classes. The table entry in row $\\chi$ and column $K$ is $\\chi_K$:\n\n\\begin{tabular}{ c | c c c }\n   & \\ldots & $K$ & \\ldots \\\\\n  \\hline\n  \\vdots &   & \\vdots & \\\\\n$\\chi$ & \\ldots   & $\\chi_K$ & \\\\\n\\vdots &    &  & \\\\\n\\end{tabular}\n\nBy convention, the first row corresponds to the trivial character, and the first column corresponds to the class of the identity, $K = \\left\\{e\\right\\}$.\n\\end{mydef}\n\nTwo equivalent procedures are provided for computing the character table of any symmetric group $S_n$ improvising from \\cite{Gillespie2012}.\n\nThe most straight forward way \\cite{sagan2013symmetric} to compute the character table is given in Algorithm ~\\ref{algo:char-tab-sagan}.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf CHARACTER-TABLE-SAGAN}}\n     \\label{algo:char-tab-sagan}\n\\begin{algorithmic}[1]\n\\Procedure {CHARACTER-TABLE-SAGAN}{$S_n$}\n\\State Determine all the partitions of $n$ which will also infer the conjugacy classes.\n\n\\State Enumerate all group elements and cluster them based on their cycle types. These clusters will coincide with the conjugacy classes.\n\n\\State For each class, compute the irreducible representation for each group element.\n\n\\State For each class, determine the character of the irreducible representation. All group elements of the same cycle type will have the same character.\n\n\\State Populate the table with the characters following the prescribed order of the partitions for both column and rows.\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nEnumeration of conjugacy classes becomes tedious when  groups larger than $S_5$ are being considered. By using the {\\it Murnaghan-Nakayama rule} , The process can be simplified even for larger groups \\cite{stanley1986enumerative} as shown by \\cite{Gillespie2012} in Algorithm ~\\ref{algo:char-tab-mn-rule}.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf CHARACTER-TABLE-GILLESPIE}}\n     \\label{algo:char-tab-mn-rule}\n\\begin{algorithmic}[1]\n\\Procedure {CHARACTER-TABLE-GILLESPIE}{$S_n$}\n\\State   The  conjugacy classes of $S_n$ are the permutations having a fixed number of cycles of each length, corresponding to a partition of $n$ called the shape of the permutation. Since, the characters of a group are constant on its conjugacy classes,  index the columns of the character table by these partitions. The partitions are arranged in an increasing order.\n\n\\State There are precisely as many irreducible characters as conjugacy classes, so the irreducible characters can be indexed by the partitions of $n$. Represent each partition as a Young diagram and write them, or the characters directly down, the left of the table in a decreasing order of the partitions.\n\n\\Comment{The Murnaghan-Nakayama Rule}\n\\State  Calculate the entry in row $\\lambda$  and column $\\mu$. Define a filling of $\\lambda$ with content $\\mu$ to be a way of writing a number in each square of $\\lambda$ such that the numbers are weakly increasing along each row and column and there are exactly $\\mu_i$ squares labeled $i$ for each $i$.\n\n\\State Consider all fillings of $\\lambda$ with content $\\mu$ such that for each label $i$, the squares labeled $i$ form a connected skew tableaux that does not contain a $2 \\times 2$ square. Such a tableaux is called a {\\it border-strip tableaux}.\n\n\\State For each label in the tableau, define the height of the corresponding border strip to be one less than the number of rows of the border strip. Weight the tableau by $\\left(-1\\right)^s$ where $s$ is the sum of the heights of the border strips that compose the tableau.\n\n\\State The entry in the character table is simply the sum of these weights.\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nNow, the concepts of hook is introduced which is used in computing characters of group representations.\n\n\\begin{mydef} [Hook]\nFor a cell $\\left(i, j\\right)$ of a Young tableau $\\lambda$, the $\\left(i, j\\right)$-hook $h_{i, j}$ is the collection of all cells of $\\lambda$ which are beneath $\\left(i, j\\right)$ (but in the same column) or t the right of $\\left(i, j\\right)$ (but in the same row), including the cell $\\left(i, j\\right)$. The {\\it length} of the hook is the number of cells appearing in the hook.\n\\end{mydef}\n\n\\begin{mydef} [Skew hook]\nA skew hook $s$ of a Young diagram $\\lambda$ is a connection collection of boundary boxes such that their removal from $\\lambda$ results in a (smaller) diagram.\n\\end{mydef}\n\nThe Murnaghan-Nakayama rule \\cite{stanley1986enumerative} is given below.\n\n\\begin{mytheo}[The Murnaghan-Nakayama rule]\n\\label{theo:mnrule}\nLet $c$ be a permutation with cycle structure $\\left(c_1, \\ldots, c_t\\right)$, $c_1 \\ge \\ldots \\ge c_t$. Then\n\n\\begin{align}\n\\chi_\\lambda \\left(c\\right) &= \\sum_{s_1, \\ldots, s_t} \\left(-1\\right)^{v\\left(s_1\\right)} \\ldots \\left(-1\\right)^{v\\left(s_t\\right)},\n\\end{align}\n\nwhere each $s_i$ is a skew hook of length $c_i$ of the diagram or partition $\\lambda$ after $s_1, \\ldots, s_{i - 1}$ have been removed, and $v \\left(s_i\\right)$ denotes the number of vertical steps in $s_i$.\n\\end{mytheo}\n\nThe semidirect product is defined as follows \\cite{dummit2004abstract}.\n\n\\begin{mydef} [Semidirect product]\nLet $H$ and $K$ be groups, with $K$ acting on $H$ via an action $\\phi : K \\to Aut \\left(H\\right)$. The multiplication operation is defined as follows.\n\\begin{align}\n\\left(h_1, k_1\\right) * \\left(h_2, k_2\\right) &= \\left(h_1 \\phi_{k_1} \\left(h_2\\right), k_1 k_2 \\right) =  \\left(h_1 \\left(k_1 \\cdot h_2\\right), k_1 k_2\\right)\n\\end{align}.\nThen the group $G$ from is called the semidirect product of $H$ by $K$, and is denoted by $H \\rtimes_\\phi K$.\n\\end{mydef}\n\n\nThe wreath product is defined as follows \\cite{dummit2004abstract}.\n\n\\begin{mydef} [Wreath product]\nLet $K$ and $L$ be groups, let $n$ be a positive integer, let $\\phi : K \\to S_n$  be a homomorphism and let $H$ be the direct product of $n$ copies of $L$. Let $\\psi$ be an injective homomorphism from $S_n$ into $Auto \\left(H\\right)$  constructed by letting the elements of $S_n$ permute the $n$ factors of $H$. The composition $\\psi \\circ \\phi$ is a homomorphism from $G$ into $Aut \\left(H\\right)$. The wreath product of $L$ by $K$ is the semidirect product $H \\rtimes K$ with respect to this homomorphism and is denoted by $L \\wr K$. \n\\end{mydef}\n\nThe dimension of an irreducible representation of the symmetric group is defined as follows.\n\n\\begin{mydef} [Dimension of an irreducible representation]\nThe dimension $dim_{\\rho_\\lambda}$ of an irreducible representation $\\rho$ for a partition $\\lambda = \\left(\\lambda_1 + \\ldots + \\lambda_i + \\ldots + \\lambda_k\\right)$ of a symmetric group $S_n$ is given as follows \\cite{fulton1991representation}.\n\n\\begin{align}\ndim_{\\rho_\\lambda} &= \\frac{n!}{l_1 \\cdot \\ldots \\cdot l_k!} \\Pi_{i < j} \\left(l_i - l_j\\right),\n\\end{align}\nwith $l_i = \\lambda_i + k - i$.\n\\end{mydef}\n\nIt is suitable to mention the following theorem on the multiplicity of an irreducible representation in the regular representation \\cite{fulton1991representation}.\n\n\\begin{mytheo}\n\\label{theo:irrep-multiplicity-in-regular}\nEvery irreducible representation $\\rho$ occurs $\\text{dim}\\left(\\rho\\right)$ times in the regular representation.\n\\end{mytheo}\n\n\\begin{proof}[Proof of Theorem ~\\ref{theo:irrep-multiplicity-in-regular}]\nLet $\\chi$ be the character of the regular representation. Then\n\\begin{align*}\n\\chi \\left(g\\right) &=\\begin{cases} \n    n       & \\quad \\text{if } g = 1, \\text{ and}\\\\\n   0  & \\quad \\text{ otherwise.}\\\\\n  \\end{cases}\n\\end{align*}\nBecause, each group elements acts by a permutation matrix, and the trace of a permutation matrix is simply the number of fixed points of the permutation. Thus,\n\\begin{align*}\n\\langle \\chi_\\rho, \\chi \\rangle &= \\frac{1}{n} \\bar{\\chi_\\rho \\left(1\\right) \\chi \\left(1\\right)}\n\\\\\n&= \\frac{1}{n} \\text{dim} \\left(\\rho\\right) n\n\\\\\n&= \\text{dim} \\left(\\rho\\right)\n\\end{align*}\n\\qedhere\n\\end{proof}\n\nAnother two important concepts in representation theory are {\\it restriction} and {\\it induction} \\cite{sagan2013symmetric}.\n\n\\begin{mydef}[Restriction]\nLet $H$ be a subgroup of $G$ and $X$ be a matrix representation of $G$. The restriction of $X$ to $H$, $X \\downarrow^G_H$, is given by\n\\begin{align*}\nX\\downarrow^G_H \\left(h\\right) = X \\left(h\\right)\n\\end{align*}\nfor all $h \\in H$.\n\\end{mydef}\n\n\\begin{mydef}[Induction]\n\\label{def:induction}\nLet $H \\le G$ and $t_1, \\ldots, t_l$ be a fixed transversal for the left cosets of $H$, i.e., $G = t_1 H \\sqcup \\ldots \\sqcup t_l H$. If $Y$ is a representation of $H$, then the corresponding induced representation $Y\\uparrow^G_H$ assigns to each $g \\in G$ the block matrix\n\\begin{align*}\nY\\uparrow^G_H \\left(g\\right) &= Y \\left(t^{-1}_i g t_j\\right)\n\\nonumber\\\\\n&= \\begin{pmatrix}\n  Y \\left(t^{-1}_1 g t_1\\right) & Y \\left(t^{-1}_1 g t_2\\right) & \\cdots & Y \\left(t^{-1}_1 g t_l\\right) \\\\\n  Y \\left(t^{-1}_2 g t_1\\right) & Y \\left(t^{-1}_2 g t_2\\right) & \\cdots & Y \\left(t^{-1}_2 g t_l\\right) \\\\\n  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n  Y \\left(t^{-1}_l g t_1\\right) & Y \\left(t^{-1}_l g t_2\\right) & \\cdots & Y \\left(t^{-1}_l g t_l\\right)\n \\end{pmatrix}\n\\end{align*}\nwhere $Y \\left(g\\right)$ is the zero matrix if $g \\notin H$.\n\\end{mydef}\n\nIt is natural to define the characters for the restricted and induced representations \\cite{serre2012linear}.\n\n\\begin{mydef}[Character of restricted representation]\nLet $X$ be a matrix representation of a group $G$, and let $H \\le G$ be a subgroup. Then, the character of the restricted representation $\\chi \\downarrow^G_H \\left(h\\right)$ is the character of the original representation $\\chi \\left(h\\right)$ for all $h \\in H$.\n\\end{mydef}\n\nThe definition of the character of induced representation is reproduced from \\cite{JohnArmstrongCharInducRepre}.\n\n\\begin{mydef}[Character of induced representation]\n\\label{def:char-ind-rep}\nLet $Y$ be a matrix representation of a group $H$ such that $H \\le G$. A transversal of $H$ in $G$ is now picked. Using the previously mentioned formula for the induced representation, it is found that,\n\\begin{align*}\n\\chi \\uparrow^G_H \\left(g\\right) &= Tr \\left( Y \\left(t^{-1}_i g t_j\\right)\\right)\n\\nonumber\\\\\n&= Tr \\begin{pmatrix}\n  Y \\left(t^{-1}_1 g t_1\\right) & Y \\left(t^{-1}_1 g t_2\\right) & \\cdots & Y \\left(t^{-1}_1 g t_l\\right) \\\\\n  Y \\left(t^{-1}_2 g t_1\\right) & Y \\left(t^{-1}_2 g t_2\\right) & \\cdots & Y \\left(t^{-1}_2 g t_l\\right) \\\\\n  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n  Y \\left(t^{-1}_l g t_1\\right) & Y \\left(t^{-1}_l g t_2\\right) & \\cdots & Y \\left(t^{-1}_l g t_l\\right)\n \\end{pmatrix}\n \\nonumber\\\\\n &= \\sum^n_{i=1} Tr \\left(Y \\left(t^{-1}_i g t_i\\right)\\right)\n \\nonumber\\\\\n &= \\sum^n_{i=1} \\chi  \\left(t^{-1}_i g t_i\\right)\n\\end{align*}\nwhere $\\chi \\left(g\\right)$ is the zero matrix if $g \\notin H$.\n\\end{mydef}\n\nSince, $\\chi$ is a class function on $H$, conjugation by any element $h \\in H$ leaves it the same. So, $\\chi \\left(h^{-1} g h\\right) = \\chi \\left(g\\right)$ for all $g \\in G$ and $h \\in H$.\n\nThe same computation is performed for each element of $H$. Then, all the results are added together and divided by the number of elements of $H$. In other words, the above function is written out in $|H|$ different ways, added  all together, and divided by $|H|$ to get exactly what the section started with started with:\n\n\\begin{align}\n\\chi \\uparrow^G_H \\left(g\\right) &= \\frac{1}{|H|} \\sum_{h \\in H} \\sum^n_{i = 1} \\chi \\left(h^{-1} t^{-1}_i g t_i h\\right)\n\\nonumber\\\\\n&= \\frac{1}{|H|} \\sum_{h \\in H} \\sum^n_{i=1} \\chi \\left(\\left(t_i h\\right)^{-1} g \\left(t_i h\\right)\\right)\n\\end{align}\nBut now as $t_i$ varies over the transversal, and as $h$ varies over $H$, their product $t_i h$ varies exactly once over $G$. That is, every $x \\in G$ can be written in exactly one way in the form $t_ h$ for some transversal element $t_i$ and subgroup element $h$. Thus, the following relation can be stated.\n\n\\begin{align}\n\\chi \\uparrow^G_H \\left(g\\right) &= \\frac{1}{|H|} \\sum_{x \\in G} \\chi \\left(x^{-1} g x\\right)\n\\end{align}\n\nIt would be appropriate if the following theorem on the {\\it Frobenius reciprocity} is also mentioned in this section \\cite{sagan2013symmetric}.\n\n\\begin{mytheo}[Frobenius reciprocity]\nLet $H \\le G$ and suppose that $\\psi$ and $\\chi$ are characters of     $H$ and $G$, respectively. Then\n\\begin{align}\n\\langle \\psi \\uparrow^G_H, \\chi \\rangle_G &= \\langle \\psi, \\chi \\downarrow^G_H \\rangle_H\n\\end{align}\nwhere the left inner product is calculated in $G$ and the right one in $H$.\n\\end{mytheo}\n\nA special case of {\\it Frobenius reciprocity} is relevant to the discussion of this paper where the representation of $H$ is the trivial representation $\\text{\\bf 1}_H$ \\cite{hallgren2000normal}. The case is described as follows.\n\n\\begin{mylem}[Special case of Frobenius reciprocity]\n\\label{lem:frob-recip-special}\nLet $H \\le G$ and suppose that $\\chi_\\rho$ is the character of the irreducible representation $\\rho$ of $G$. Then\n\\begin{align}\n\\langle {\\chi \\uparrow^G_H}_{\\text{\\bf 1}_H}, \\chi_\\rho \\rangle_G &= \\langle \\chi_{\\text{\\bf 1}_H}, \\chi_\\rho \\downarrow^G_H \\rangle_H\n\\end{align}\nwhere the left inner product is calculated in $G$ and the right one in $H$.\n\\end{mylem}\n\nThe Example $3.13$ from  \\cite{fulton1991representation} can be reproduced here in relevance to the ongoing discussion.\n\n\\begin{myrem}\n\\label{rem:triv-rep-induc}\nThe permutation representation associated to the left action of $G$ on $G\/H$ is induced from the trivial one-dimensional representation $W$ of $H$. Here, the representation of $G$, $V$ has basis $\\left\\{e_\\sigma: \\sigma \\in G\/H\\right\\}$, and $W = \\mathbb{C}\\cdot e_H$, with     $H$the trivial coset.\n\\end{myrem}\n\nFollowing Remark ~\\ref{rem:triv-rep-induc}, it can be said that $\\text{\\bf 1}_H\\uparrow^G_H $ is the permutation representation of $G$. So, according to the Theorem ~\\ref{theo:irrep-multiplicity-in-regular}, the multiplicity of $\\rho$ in $\\text{\\bf 1}_H\\uparrow^G_H $ is $d_\\rho$.\n\n\\subsection{Quantum Fourier Sampling}\nQuantum Fourier Sampling is a class of quantum algorithms which uses quantum Fourier transformation as a subroutine.\nThe definition of quantum Fourier transform of a map from a finite group to its representation is defined as follows where the notations are borrowed from \\cite{hallgren2000normal}. Later in this section, the algorithm is also presented.\n\n\\begin{mydef}[Fourier transformation of a finite group]\n\\label{def:qft}\nLet $f: G \\to \\mathbb{C}$. The Fourier transform of $f$ at the irreducible representation $\\rho$ is the $d_\\rho \\times d_\\rho$ matrix\n\\begin{align}\n\\hat{f} \\left(\\rho\\right) &= \\sqrt{\\frac{d_\\rho}{|G|}} \\sum_{g\\in G} f\\left(g\\right) \\rho \\left(g\\right)\n\\end{align}\n\\end{mydef}\n\nIn quantum Fourier transform, the superposition $\\sum_{g\\in G} f_g |g\\rangle$ is identified  with the function $f: G \\to \\mathbb{C}$ defined by $f \\left(f \\left(g\\right)\\right) = f_g$. Using this notation, $\\sum_{g\\in G} f\\left(g\\right) |g\\rangle$ is mapped under the Fourier transform to $\\sum_{\\rho \\in \\hat{G}, 1 \\le i, j\\le d_\\rho} \\hat{f}\\left(\\rho\\right)_{i, j}|\\rho, i, j\\rangle$. Here, $\\hat{G}$ is the set of all irreducible representations of $G$ and $\\hat{f}\\left(\\rho\\right)_{i,j}$ is a complex number. The probability of measuring the register $|\\rho \\rangle$ is\n\\begin{align}\n\\sum_{1\\le i, j\\le d_\\rho} |\\hat{f}\\left(\\rho\\right)_{i,j}|^2 &= \\|\\hat{f}\\left(\\rho\\right)\\|^2\n\\end{align}\nwhere $\\|A\\|$ is the natural norm (also known as Frobenius norm) given by $\\|A\\|^2 = Tr \\left(A^\\dagger A\\right)$.\n\nThe Frobenius norm can be calculated from the characters of the group associated which is demonstrated in the following theorem reproduced from \\cite{hallgren2000normal}.\n\n\\begin{mytheo}\n\\label{theo:frob-char}\nIf, $f$ is an indicator function of a left closet of $H$ in $G$, i.e. for some $c \\in G$,\n\\begin{align}\nf \\left(g\\right) &= \\begin{cases}\n    \\frac{1}{\\sqrt{|H|}}       & \\quad \\text{ if } g \\in c H, \\text{ and }\\\\\n    0  & \\quad 0 \\text{ otherwise}\\\\\n  \\end{cases}\n\\end{align}, then,\n\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\frac{|H|}{|G|} d_\\rho \\langle \\chi_\\rho, \\chi_{\\text{\\bf 1}_H} \\rangle_H\n\\end{align}\n\\end{mytheo}\n\n\\begin{proof}[Proof of Theorem ~\\ref{theo:frob-char}]\nFrom Definition ~\\ref{def:qft}, it is known that,\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\sum_{1\\le i, j\\le d_\\rho} |\\hat{f}\\left(\\rho\\right)_{i,j}|^2\n\\end{align}\nOnly  $\\rho$ is to be measured. \n\nFollowing relation is assumed in the theorem.\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\| \\rho \\left(c\\right) \\sum_{h\\in H} \\rho \\left(h\\right) \\|^2\n\\end{align}\n$\\rho \\left(c\\right)$ is a unitary matrix. So, as a multiplier it does not change the norm \\cite{meyer2000matrix}.\n\\begin{align}\n\\|\\hat{f} \\left(\\rho\\right)\\|^2 &= \\| \\sum_{h\\in H} \\rho \\left(h\\right) \\|^2\n\\end{align}\n\nSo, the probability of measuring $\\rho$ is determined by $\\sum_{h\\in H} \\rho \\left(h\\right)$. If correctly normalized, $\\frac{1}{|H|}\\sum_{h\\in H}\\rho(h)$ is a projection.\n\n\\begin{align}\n\\left(\\frac{1}{|H|}\\sum_{h\\in H}\\rho(h)\\right)^2&=\\frac{1}{|H|^2}\\sum_{h_1,h_2\\in H} \\rho(h_1 h_2)\n\\nonumber\\\\\n&=\\frac{1}{|H|}\\sum_{h\\in H}\\rho(h)\n\\end{align}\nbecause $h_1h_2=h$ has $|H|$ solutions $(h_1,h_2)\\in H\\times H$.\n\n With the right choice of basis,   $\\hat{f} \\left(\\rho\\right)$ will be diagonal and consist of ones and zeros. The probability of measuring $\\rho$ will then be the sum of ones in the diagonal. As $\\rho$ is an irreducible representation of $G$,  the sum of the matrices $\\rho \\left(h\\right)$ for all $h \\in H$ needs to be taken into account. Based of the assumption of the current theorem, one may only consider to evaluate $\\rho$ on $H$. According to the assumption, the probability of measuring $\\rho$ when $g \\notin cH$ is zero. So, one may consider consider $\\rho \\downarrow^G_H$ instead of $G$.\n\nThen, the Fourier transform of $f$ at $\\rho$ is comprised of blocks, each corresponding to a representation in the decomposition of $\\rho\\downarrow^G_H$. Such as,\n\\begin{align}\n\\sum_{h \\in H} \\rho \\left(h\\right) &= U  \\begin{bmatrix}\n\\sum_{h \\in H} \\sigma_1 \\left(h\\right) & 0 & \\cdots & 0 \\\\\n  0 & \\sum_{h \\in H} \\sigma_2 \\left(h\\right) & \\cdots & 0 \\\\\n  \\vdots  & \\vdots  & \\ddots & \\vdots  \\\\\n 0 & 0 & \\cdots & \\sum_{h \\in H} \\sigma_t \\left(h\\right)\n \\end{bmatrix} U^\\dagger\n\\end{align}\nHere, $U$ is an arbitrary unitary transformation, $\\sigma_i$ is an irreducible representation of $H$ with possible repetition. Now, as a special case of the orthogonality relation among group characters,  $\\sum_{h \\in H} \\rho \\left(h\\right)$ is nonzero only when the irreducible representation is trivial, in which case, it is $|H|$.\n\nSo, the probability of measuring $\\rho$ is:\n\\begin{align}\n\\norm{\\hat{f} \\left(\\rho\\right)}^2 &= \\norm{ \\sqrt{\\frac{d_\\rho}{|G|}} \\sum_{h\\in H} \\rho \\left(h\\right) }^2\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|}  \\norm{ \\sum_{h\\in H} \\rho \\left(h\\right) }^2\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|} \\mathrm{tr}\\left(\\sum_{h_1\\in H}\\rho(h_1)\\right)\\left(\\sum_{h_2\\in H}\\rho(h_2)\\right)^\\dagger\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|} \\mathrm{tr}\\sum_{h_1,h_2\\in H}\\rho(h_1h_2^{-1})\n\\nonumber\\\\\n&= \\frac{d_\\rho}{|G|} \\frac{1}{|H|} |H|^2 \\langle \\chi_\\rho, \\chi_{\\text{\\bf 1}_H}\\rangle_H\n\\nonumber\\\\\n&= \\frac{|H|}{|G|} d_\\rho \\langle \\chi_\\rho, \\chi_{\\text{\\bf 1}_H} \\rangle_H\n\\end{align}\nIt should be mentioned that, by definition, $\\rho$ appears  $\\langle \\chi_\\rho, \\chi_{1_H} \\rangle_H$ times in the decomposition of $\\text{\\bf 1}_H$.\n\\qedhere\n\\end{proof}\n\nInterested readers are encouraged to refer to \\cite{diaconis1990efficient} for a review on the classical complexity of Fourier transformation of the symmetric groups. The goal of quantum Fourier sampling algorithm is to sample the labels and elements of the irreducible representations available after quantum Fourier transformation. Sampling only the labels of representations is called weak sampling. On the other hand, sampling also the indices of the elements of the matrix is called strong Fourier sampling. The quantum Fourier sampling algorithm is reproduced from \\cite{hallgren2000normal}.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf QUANTUM-FOURIER-SAMPLING}}\n \\label{algo:weak}\n\\begin{algorithmic}[1]\n\\Procedure {QUANTUM-FOURIER-SAMPLING}{$f : G \\to S$}\n\\State Compute $\\sum_{g \\in G} |g, f \\left(g\\right) \\rangle$ and measure the second register $f \\left(g\\right)$. The resulting super-position is $\\sum_{h \\in H} |c h \\rangle \\otimes | f \\left(c h\\right)\\rangle$ for some coset $c H$ of $H$. Furthermore, $c$ is uniformly distributed over $G$.\\;\n\n\\State  Compute the Fourier transform of the coset state which is $\\sum_{\\rho \\in \\hat{G}} \\sqrt{\\frac{d_\\rho}{|G|}} \\sqrt{\\frac{1}{|H|}} \\left(\\sum_{h \\in H} \\rho \\left(c h\\right)\\right)_{i, j} | \\rho, i, j\\rangle$, where $\\hat{G}$ denotes the set of irreducible representations of $G$.\\;\n\n\\State  Measure the first register and observe a representation $\\rho$ (weak) or $\\rho, i, \\text{ and } j$ (strong). \\;\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Quantum Fourier sampling for graph isomorphism}\n\\label{sec:qft-gi}\nThe goal of this section is to give the readers a detailed exposition of the standard approach of hidden subgroup algorithms for graph isomorphism problems. In this section, the quantum Fourier sampling algorithm for graph isomorphism is described. Although it is reproduced verbatim from \\cite{hallgren2000normal}, the notation is changed to match it to this paper's original discussion of quantum Fourier sampling for graph automorphism. Most of the algorithms for the non-Abelian hidden subgroup problem use a black box for $\\varphi$ in the same way as in the Abelian hidden subgroup problem \\cite{lomonacopers2002}. This has come to be known as the {\\it standard method}. The standard method begins by preparing a uniform superposition over group elements \\cite{childs2016lecture}:\n\n\\begin{align}\n|S_n \\rangle := \\frac{1}{\\sqrt{|S_n|}} \\sum_{g\\in S_n} |g\\rangle\n\\end{align}\n\nThe value of $\\varphi \\left(g\\right)$ is then computed in an ancilla register which creates the following state.\n\n\\begin{align}\n \\frac{1}{\\sqrt{|S_n|}} \\sum_{g\\in S_n} |g, \\varphi \\left(g\\right) \\rangle\n\\end{align}\n\nThen  the second register is discarded by just being traced it out. If the outcome of the second register is $s$  then the state is projected onto the uniform superposition of those $g \\in S_n$ such that $\\varphi \\left(g\\right) = s$. By definition of $\\varphi$, it is some left coset of the hidden subgroup $H$. Since every coset contains the same number of elements, each left coset occurs with equal probability. Thus, the standard method produces the following coset state.\n\n\\begin{align}\n|g H\\rangle := \\frac{1}{\\sqrt{|H|}} \\sum_{h \\in H} |g h \\rangle\n\\end{align}\n\nor equivalently as the following mixed {\\it hidden subgroup state}.\n\n\\begin{align}\n\\rho_H := \\frac{1}{|S_n|} \\sum_{g \\in S_n} |g h \\rangle \\langle g h |\n\\end{align}\n\nIt has been previously mentioned that $\\varphi$ maps the group elements of $S_n$ to $\\text{Mat}\\left(\\mathbb{C}, N \\right)$. Here,  more information is presented about the space $\\text{Mat}\\left(\\mathbb{C}, N \\right)$. Let the complete set of  irreducible representations of $S_n$ (which are unique up to isomorphism) be $\\hat{S_n}$. The Fourier transform is a unitary transformation from the group algebra, $\\mathbb{C} S_n$, to a complex vector space whose basis vectors correspond to matrix elements of the irreducible representations of $S_n$, $\\oplus_{\\rho \\in \\hat{S_n}} \\left(\\mathbb{C}^{d_\\rho} \\otimes \\mathbb{C}^{d_\\rho}\\right)$. Here, $d_\\rho$ is the dimension of the irreducible representation $\\rho$. \n\n$|g \\rangle$ is the basis vector chosen for the group element $g \\in S_n$. There will be $n !$ such basis vectors of dimension $n! \\times 1$. For a given group element $g$, there are a particular number of matrices  one for each irreducible representation $\\rho$. $|g h\\rangle$ is expressed as $|\\rho, j, k\\rangle$ which is the basis vector labeled by the $\\left( j, k \\right)$-th element of the irreducible representation $\\rho$ of $g$.\n\nAs it is mentioned earlier, when only $\\rho$ is measured  from $|\\rho, j, k\\rangle$, it is called {\\it weak Fourier sampling}. In {\\it strong Fourier sampling}, $j$ and $k$ are also measured.\n\n\\subsection{Weak Fourier sampling for $\\text{{\\bf GI}}_{\\text{HSP}}$}\n\\label{sec:wfs-gi}\nThis section summarizes what already is known about the weak Fourier sampling when applied to the graph isomorphism problem. The weak Fourier sampling for $\\text{{\\bf GI}}_{\\text{HSP}}$ attempts to measure the labels of irreducible representations of the symmetric group $S_{2 n}$ when the input graphs $\\Gamma_1$ and $\\Gamma_2$ are of $n$ vertices. It is assumed that $Aut \\left(\\Gamma_1\\right) = Aut \\left(\\Gamma_2\\right) = \\left\\{e\\right\\}$. If $\\Gamma = \\Gamma_1 \\sqcup \\Gamma_2$, one of the following two claims is true \\cite{hallgren2000normal}. \n\n\\begin{itemize}\n\\item If $\\Gamma_1 \\not\\approx \\Gamma_2$, then $\\text{Aut}\\left(\\Gamma \\right) = \\left\\{e\\right\\}$.\n\\item If $\\Gamma_1 \\approx \\Gamma_2$, then $\\text{Aut}\\left(\\Gamma \\right) = \\left\\{e, \\sigma\\right\\} = \\mathbb{Z}_2$, where $\\sigma \\in S_{2n}$ is a permutation with $n$ disjoint $2$-cycles.\n\\end{itemize}\n\nIt should be mentioned that in \\cite{hallgren2000normal}, the authors derived the success probability of measuring the label of the irreducible representations for $S_n$. In this paper, the same probability will be derived for $S_{2 n}$ to keep consistency with the definition of $\\text{{\\bf GI}}_{\\text{HSP}}$.\n\nThe weak Fourier sampling algorithm for finding $\\text{Aut}(\\Gamma)$ in $S_{2 n}$ is described below.\n\n\\begin{algorithm}[H]\n\\caption{{\\bf WEAK-QUANTUM-FOURIER-SAMPLING-}$S_{2 n}$}\n \\label{algo:weak-gi}\n\\begin{algorithmic}[1]\n\\Procedure {WEAK-QUANTUM-FOURIER-SAMPLING-$S_{2 n}$}{a graph $\\Gamma$ such that either $\\text{Aut} \\left(\\Gamma\\right) = \\left\\{e\\right\\}$ or $\\text{Aut} \\left(\\Gamma\\right) = \\left\\{e, \\sigma\\right\\}$}\n\\State    Compute $ \\frac{1}{\\sqrt{\\left(2 n\\right)!}} \\sum_{g\\in S_{2 n}} |g, \\varphi \\left(g\\right) \\rangle$\\;\n\n\\State  Compute $\\sum_{\\rho \\in \\hat{S_{2 n}}} \\sqrt{\\frac{d_\\rho}{\\left(2 n\\right)!}} \\sqrt{\\frac{1}{|\\text{Aut}\\left(\\Gamma \\right)|}} \\left(\\sum_{h \\in \\text{Aut}\\left(\\Gamma \\right)} \\rho \\left(c h\\right)\\right)_{i, j} |\\rho, i, j \\rangle $\\;\n\n\\State  Measure $\\rho$ as in tracing it out \\;\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\nLet $p_\\rho$ be the probability of sampling $\\rho$ in Algorithm ~\\ref{algo:weak-gi} when $\\Gamma_1 \\not\\approx \\Gamma_2$, and $q_\\rho$ when $\\Gamma_1 \\approx \\Gamma_2$. So,  the induced representation of $\\text{Aut} \\left(\\Gamma\\right)$ to $S_{2 n}$, $\\text{Ind}^{S_{2 n}}_{\\text{Aut} \\left(\\Gamma\\right)} 1$, is the regular representation. So, $\\langle \\chi_\\rho | \\chi_{\\text{Ind}^{S_{2 n}}_{\\text{Aut} \\left(\\Gamma\\right)} 1} \\rangle$, the multiplicity of $\\rho$ in the regular representation, is $d_\\rho$. Hence, $p_\\rho = \\frac{d^2_\\rho}{\\left(2 n\\right)!}$.\n\nWhen $\\Gamma_1 \\approx \\Gamma_2$, $\\text{Aut} \\left(\\Gamma\\right) = \\left\\{e, \\sigma\\right\\}$. In this case, the probability of measuring $\\rho$, \n\n\\begin{align}\nq_\\rho &= \\frac{|\\text{Aut}\\left(\\Gamma \\right)|}{\\left(2 n\\right)!} d_\\rho \\langle \\chi_1 | \\chi_\\rho \\rangle_{\\text{Aut}\\left(\\Gamma \\right)}\n\\end{align}\n\n$\\text{Aut}\\left(\\Gamma \\right)$ has only two elements, $e$ and $\\sigma$, hence\n\n\\begin{align}\n \\langle \\chi_1 | \\chi_\\rho \\rangle_{\\text{Aut}\\left(\\Gamma \\right)} &= \\frac{1}{2} \\left(\\chi_\\rho \\left(e\\right) + \\chi_\\rho \\left(\\sigma\\right)\\right)\n \\nonumber\\\\\n &= \\frac{1}{2} \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n\\end{align}\n\nSo, \n\n\\begin{align}\nq_\\rho &= \\frac{|\\text{Aut}\\left(\\Gamma \\right)|}{\\left(2 n\\right)!} d_\\rho \\frac{1}{2} \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n \\nonumber\\\\\n &= \\frac{2}{\\left(2 n\\right)!} d_\\rho \\frac{1}{2} \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n \\nonumber\\\\\n &=\\frac{d_\\rho}{\\left(2 n\\right)!}  \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)\n\\end{align}\n\nSo, \n\n\\begin{align}\n| p_\\rho - q_\\rho| &=| \\frac{d^2_\\rho}{\\left(2 n\\right)!} - \\frac{d_\\rho}{\\left(2 n\\right)!}  \\left(d_\\rho + \\chi_\\rho \\left(\\sigma\\right)\\right)|\n \\nonumber\\\\\n &=  \\frac{d_\\rho}{\\left(2 n\\right)!}   |\\chi_\\rho \\left(\\sigma\\right)|\n\\end{align}\n\nNow, the Murnaghan-Nakayama rule (Theorem ~\\ref{theo:mnrule}) is used to approximate  $|\\chi_\\rho \\left(\\sigma\\right)|$. \n\nThe number of unordered decompositions for the diagram $\\lambda$ with $2 n$ cells is $2^{4 n} = 16^n$. For each unordered decomposition, the number of ordered decomposition is at most $\\left(2 \\sqrt{2 n}\\right)^{\\frac{2 n}{2}} = \\left(2 \\sqrt{2 n}\\right)^{n}$ . So, by the Murnaghan-Nakayama rule, $\\chi_\\rho \\left(\\sigma\\right) \\le 16^n \\left(2 \\sqrt{2 n}\\right)^{n}$ .\n\nSo,\n\n\\begin{align}\n|\\chi_\\rho \\left(\\sigma\\right)| & \\le 16^n \\left(2 \\sqrt{2 n}\\right)^{n}\n\\nonumber\\\\\n &\\le 2^{4 n} 2^{n} \\sqrt{2}^n \\left( \\sqrt{ n}\\right)^{n}\n \\nonumber\\\\\n &\\le 2^{O\\left( n\\right)}  n^{\\frac{n}{2}}\n\\end{align}\n\nNow $| p_\\rho - q_\\rho|$ is computed for all irreducible representations. So,\n\n\\begin{align}\n| p - q|_1 &= \\sum_\\rho | p_\\rho - q_\\rho|\n \\nonumber\\\\\n & \\le \\sum_\\rho \\frac{d_\\rho}{\\left(2 n\\right)!} 2^{O\\left( n\\right)}  n^{\\frac{n}{2}}\n  \\nonumber\\\\\n  & \\le \\sum_\\rho \\frac{\\sqrt{\\left(2n\\right)!}}{\\left(2 n\\right)!} 2^{O\\left( n\\right)}  n^{\\frac{n}{2}}\n\\nonumber\\\\\n&= \\frac{2^{O\\left( n\\right)}  }{  \\frac{n}{2}^{\\frac{n}{2}}}\n\\nonumber\\\\\n&\\le \\frac{2^{O\\left( n\\right)}  }{  \\left(\\frac{n}{2}\\right)!}\n\\nonumber\\\\\n&\\lll 2^{-\\Omega \\left(n\\right)}\n\\end{align}\nSo, the probability of successfully measuring the labels of the irreducible representations is exponentially low in the size of the graphs.\n\n\\section{Quantum Fourier sampling for cycle graph automorphism}\n\\label{sec:weak-cycle}\nIn this section, it is shown that quantum Fourier sampling fails to compute the automorphism group of a cycle graph. The result is original to this paper. The scheme of the proof is as follows. First,  the automorphism group of the graph is computed which is trivial for this case. Then  its irreducible representation is computed. Then the general expressions for group characters are derived. And, finally, using the characters, the probabilities of measuring the labels of irreducible representations are computed. A more granular representation of the previously mentioned steps is given below.\n\n            \\begin{tikzpicture}[node distance=5cm, scale=0.8, every node\/.append style={transform shape}]\n\n\\node[text width=4cm] (tf) [startstop] {\\scriptsize Determine the symmetric group in which\\\\ the automorphism group is hidden in};\\\\\n\n\\node[text width=4cm] (ln) [startstop , right of = tf] {\\scriptsize Determine the presentation of the\\\\ automorphism group};\n\n\\node[text width=4cm] (mv) [startstop , below of = ln] {\\scriptsize Determine the order of the\\\\ automorphism group};\n\n\\node[text width=4cm] (qp) [startstop , left of = mv] {\\scriptsize Determine the irreducible representations\\\\ of the automorphism group};\n\n\\node[text width=4cm] (bq) [startstop , below of = qp] {\\scriptsize Determine the characters\\\\ of the representations};\n\n\\node[text width=5cm] (bq1) [startstop , right of = bq] {\\scriptsize Compute the inner product of the characters of the trivial representation of the automorphism group and the irreducible representations of the symmetric group restricted to the automorphism group};\n\n\\node[text width=5cm] (bq2) [startstop , below of = bq1] {\\scriptsize To compute previous quantity, compute the inner product of the characters of the trivial representation of the automorphism group induced up to the symmetric group and the irreducible representation of the symmetric group};\n\n\\node[text width=4cm] (bq3) [startstop , left of = bq2] {\\scriptsize Using the previous quantity compute the probability of sampling the labels of the irreducible representations of the automorphism group};\n\n\\draw [arrow] (tf) -- (ln);\n\\draw [arrow] (ln) -- (mv);\n\\draw [arrow] (mv) -- (qp);\n\\draw [arrow] (qp) -- (bq);\n\\draw [arrow] (bq) -- (bq1);\n\\draw [arrow] (bq1) -- (bq2);\n\\draw [arrow] (bq2) -- (bq3);\n\n\\end{tikzpicture}\n\n\\subsection{Automorphism group of cycle graph}\nThe automorphism group of an $n$-cycle graphs is the dihedral group $D_n$ of order $2 n$. This section intends to study the weak Fourier sampling of cycle graphs. To provide the background, this section discusses the irreducible representations of the dihedral group $D_n$. \n\n\\begin{mydef}[Cycle Graph]\nAn $n$-cycle graph is a single cycle with $n$ vertices.\n\\end{mydef}\n\nThe automorphism group of an $n$-cycle graph is the dihedral group $D_n$ which is of order $2 n$.  If $D_n$ is even, the group can be generated as $\\langle(2\\quad n)(3 \\quad n-1) \\ldots (\\frac{n}{2}-1 \\quad \\frac{n}{2}+1),\\, (1 \\ldots n)\\rangle$. If $D_n$ is odd, the group can be generated as $\\langle(2 \\quad n) (3 \\quad n-1) \\ldots (\\frac{n-1}{2} \\quad \\frac{n+1}{2}),\\, (1 \\ldots n)\\rangle$. This is a manifestation of the presentation \n$D_n = \\langle x, y \\mid x^n = y^2 = (xy)^2 = 1, y x y=x^{-1}\\rangle$. The correspondence consists of $x = (1 \\ldots n)$ and  $y = (2 \\quad n) (3 \\quad n-1) \\ldots (\\frac{n}{2}-1\\quad\\frac{n}{2}+1)$ if $n$ is even, and  $y = (2 \\quad n) (3 \\hspace{0.5cm} n-1) \\ldots (\\frac{n-1}{2} \\quad\\frac{n+1}{2})$ if $n$ is odd. The orders of $x$ and $y$ are $n$ and $2$ respectively.\n\nThe order of the symmetric group $S_n$ is $n!$. The order of a dihedral group $D_n$ is $2 n$.  So, the index of $D_n$ in $S_n$ is $\\frac{n!}{2n} \\approx \\frac{\\sqrt{2 \\pi n} \\left(\\frac{n}\n{e}\\right)^n}{2n}$.\n\nIt would be relevant if the following  important theorem proved in \\cite{conrad2009dihedral} is mentioned here.\n\n\\begin{mytheo}\n\\label{theo:d-n-subgroups}\nEvery subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups is as follows:\n\\begin{itemize}\n\\item $\\langle x^d \\rangle$, where $d \\mid n$, with index $2 d$,\n\\item $\\langle x^d, x^i y \\rangle$, where $d \\mid n$ and $0 \\le i \\le d-1$, with index $d$.\n\\end{itemize}\nEvery subgroup of $D_n$ occurs exactly once in this listing.\n\nIn this theorem, subgroups of the first type are cyclic and subgroups of the second type are dihedral: $\\langle x^d \\rangle \\cong \\text{\\bf Z}\/\\left(n\/d\\right)$ and $\\langle x^d, x^i y  \\rangle \\cong D_{n\/d}$.\n\\end{mytheo}\n\nBased on Theorem ~\\ref{theo:d-n-subgroups}, following remark can be made.\n\n\\begin{myrem}\n\\label{rem:d-n-subgroup-order}\nThe order of the subgroups $\\langle x^d \\rangle$ and $\\langle x^d, x^i y \\rangle$ are $\\frac{n}{d}$ and $\\frac{2 n}{d}$ respectively.\n\\end{myrem}\n\nEvery element of $D_n$ is either $x^i$ or $y x^i$ for $0 \\le i < n$. The conjugacy classes are small enough in number to be enumerated.\n\n$\\begin{array}{rlclcl}\n\\text{Conjugate} &x^i &\\text{by}&x^j &:&(x^j) x^i(x^{-j})=x^i\\\\\n                 &x^i &\\text{by}&yx^j&:&(yx^j)x^i(x^{-j}y)= yx^iy=x^{-i}\\\\\n                 &yr^i&\\text{by}&r^j &:&(r^j) yr^i(r^{-j}) = yr^{-j}r^ir^{-j} = yr^{i-2j}\\\\\n                 &yx^i&\\text{by}&yx^j&:&(yx^j)yx^i(x^{-j}y) = x^{i-2j}y=yx^{2j-i}\\\\\n\\end{array}$\n\nThe set of rotations decomposes into inverse pairs, $\\left\\{x^i, \\left(x^i\\right)^{-1}\\right\\}$. So, the classes are $\\left\\{1\\right\\}, \\left\\{x, x^{n-1}\\right\\}$, $\\left\\{x^2, x^{n-2}\\right\\}, \\ldots$. When $n$ is even, there are $\\frac{n}{2}+1$, and when $n$ is odd, there are $\\frac{n+1}{2}$ conjugacy classes.\n\n$y x$ is conjugate to $y x^3, y x^5, \\ldots$ while $y$ is conjugate to $y x^2, y x^4, \\ldots$. If $n$ is even, these two sets are disjoint. However, $y x$ is conjugate to $y x^{n-1}$ (via $x$), so if $n$ is odd, all the non trivial reflections are in one conjugacy class.\n\nSo, the total number of conjugacy classes are as follows. If $n$ is even, the total number of conjugacy classes is $\\left(\\frac{n}{2}+1\\right) + 2 = \\frac{n}{2}+3$. If $n$ is odd, the total number of conjugacy classes is $\\frac{n+1}{2} + 1 = \\frac{n+3}{2} $.\n\nThe commutators of $D_n$,\n\n\\begin{align}\n\\left[x^i, y x^j\\right] &=  x^{-i}\\left(y x^j\\right)x^i\\left(y x^j\\right)\n\\nonumber\\\\\n& = y x^{2i+j}y x^j\n\\nonumber\\\\\n& = \\left(x^i\\right)^2\n\\end{align}\n\n\\subsection{Irreducible representations}\n\\label{sec:irrepdn}\nThe commutators generate the subgroup of squares of rotations. When $n$ is even, only half the rotations are squares, hence $G\/\\left[G, G\\right]$ is of order four. When $n$ is odd, all rotations are squares, hence $G\/\\left[G, G\\right]$ is of order two. The number of one dimensional irreducible representations is the order of $G\/\\left[G, G\\right]$. So,  when $n$ is even, there are four one dimensional representations and when $n$ is odd, there are two one dimensional representations.\n\nThe representations can be enumerated as follows.\n\n\\begin{itemize}\n\\item When $n$ is even:\n\\begin{itemize}\n\\item The trivial representation, sending all group elements to the $1 \\times 1$ matrix $\\begin{pmatrix}1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x^2, y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x^2, x y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\end{itemize}\n\\item When $n$ is odd:\n\\begin{itemize}\n\\item The trivial representation, sending all group elements to the $1 \\times 1$ matrix $\\begin{pmatrix}1\\end{pmatrix}$.\n\\item The representation, sending all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\\end{itemize}\n\\end{itemize}\n\nThe two dimensional irreducible representations are described as follows. There is an obvious subgroup $\\{1, x, \\ldots, x^{n-1}\\}$ which is a cyclic group of order $n$. It can be defined as $C_n < D_{n}$. Since $C_n$ is abelian, it has $n$ irreducible 1-dimensional representations over $\\mathbb{C}$, namely\n\n\\begin{align}\nx\\mapsto e^{2\\pi ki\/n},\\qquad 0 &\\leq k < n\n\\end{align}\n\nwhich captures the idea of rotating by an angle of $2\\pi k\/n$. These easily-described representations are induced to $D_{n}$ in order to find some possibly new representations.\n\nFor the representation $W$ of a subgroup $H\\leq G$ (i.e. an H-linear action on $W$), the induced representation of $W$ is \n\n\\begin{align}\n\\bigoplus_{g\\in G\/H}g\\cdot W\n\\end{align}\n\nwhere $g$ ranges over a set of representatives of $G\/H$.\n\nThe induced representation of $C_n$ to $D_{n}$ for fixed $k$ is straight forward since $D_{n}\/C_n$ has representatives $\\{1, y\\}$. So, one just need to describe the $D_{n}$-vector space $\\mathbb{C}\\oplus y \\cdot\\mathbb{C}$ where $\\mathbb{C}$ has basis consisting only of $w_1$. Now, the $D_{n}$ action turns into an actual matrix representation.\n\nSpecifically, it can be found out how $x$ acts on each summand using the representation of $C_n$: \n\n\\begin{align}\nx\\cdot w_1 &= e^{2\\pi ki\/n}w_1, \\text{ and}\n\\end{align}\n\n\\begin{align}\nx \\cdot(y \\cdot w_1) &= x y \\cdot w_1 \n\\nonumber\\\\\n&= y x^{-1}\\cdot w_1 = e^{-2\\pi ki\/n} y \\cdot w_1\n\\end{align}\n\n which means $x$ acts by the matrix \n $\\begin{pmatrix}\n e^{2\\pi ki\/n}&0\\\\\n 0&e^{-2\\pi ki\/n}\n \\end{pmatrix}\n  $.\n\nIt can also be figured out how $y$ acts. $y$ obviously takes $w_1$ to $y \\cdot w_1$, and $y$ takes $y \\cdot w_1$ to $y^2 w_1=w_1$, so $y$ simply interchanges the two summands. This entails  that $y$ acts by the matrix \n$\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n$.\n\nHere,  the $k$-th two dimensional irreducible representations are listed for the general group elements.\n\n\\begin{align}\nx \\mapsto \\begin{pmatrix}\ne^{\\frac{2 \\pi i k}{n}}&0\\\\\n0&e^{-\\frac{2 \\pi i k}{n}}\n\\end{pmatrix}\n\\nonumber\\\\\nx^l \\mapsto \\begin{pmatrix}\ne^{\\frac{2 \\pi i k l}{n}}&0\\\\\n0&e^{-\\frac{2 \\pi i k l}{n}}\n\\end{pmatrix}\n\\nonumber\\\\\ny \\mapsto \\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n\\nonumber\\\\\nx^l y \\mapsto \\begin{pmatrix}\n0&e^{\\frac{2 \\pi i k l}{n}}\\\\\ne^{-\\frac{2 \\pi i k l}{n}}&0\n\\end{pmatrix}\n\\end{align}\n\nIt is observed that, $0 \\le l \\le n-1$. Both $l = 0$ and $l = n$ determine the identity matrix to which the identity element, $e$, is mapped. When $n$ is even, the $k$-th and $\\left(n-k\\right)$-th representations are equivalent, hence the distinct representations are found only for $k = 1, 2, \\ldots, \\frac{n-2}{2}$. The representations for $k = 0$ and $k = \\frac{n}{2}$ are not irreducible and they decompose into one dimensional representations. On the other hand, when, $n$ is odd, there are $\\frac{n-1}{2}$ irreducible representations.\n\nUsing the previous calculations,  the total number of irreducible representations for $D_n$ can be computed. When $n$ is even, the total number is $\\frac{n-2}{2} + 4 = \\frac{n}{2} + 3$. When $n$ is odd, it is $\\frac{n-1}{2} + 2 = \\frac{n+3}{2}$. \n\nAt this point, the following remark regarding the characters of the irreducible two dimensional representations can be made. \n\n\\begin{myrem}\n\\label{rem:char-dn}\nThe characters of the representations of the elements of type $y$ and $x^l y$ are both zeros. The representations of the elements of type $x$ have the same character, $2 \\cos \\left(\\frac{2 \\pi k}{n}\\right)$. Finally, the representations of the elements of type $x^l$ have the same character, $2 \\cos \\left(\\frac{2 \\pi k l}{n}\\right)$.\n\\end{myrem}\n\n\\subsection{Sampling the representations}\nThe interest of this paper with the dihedral group $D_n$, of order $2 n$, is based on the fact that it is the automorphism group of the $n$-cycle graph. A $p$-group is a group where the order of every group element is a power of the prime $p$.  So, $D_n$ can be a $p$-group only when $n = 2^m$, when $m \\in \\mathbb{Z}$, because that is when $x^{2^m} = y^{2^1} = 1$ for $p = 2$. \n\nThe restrictions from the irreducible representations of $S_n$ to $D_n$ is straight forward. For any irreducible representation $\\rho$ of $S_n$, its restriction to $D_n$ is $\\rho \\left(h\\right)$ for all $h \\in D_n$. This new representation may not be necessarily irreducible.\n\nThe inductions from the irreducible representations of $D_n$ to $S_n$ can be computed following the Definition ~\\ref{def:induction}. This new representation may also not be necessarily irreducible. \n\nFollowing \\cite{hallgren2000normal}, this paper seeks to compute the induced representation of  $\\text{\\bf 1}_{D_n}$ which is the trivial representation of the dihedral group $D_n$. As a prerequisite,  a transversal for the left cosets of $D_n$ needs to be computed which can be done by any of the two classical algorithms presented in Section $4.6.7$ of \\cite{holt2005handbook}. The computation of  a transversal of a group requires the computation of the base of a group which can be computed in polynomial time using the Schreier-Sims algorithm \\cite{seress2003permutation, sims1970computational, knuth1991efficient}.\n\nAs it has been mentioned previously in the current section, there are $l = \\frac{n!}{2n} \\approx \\frac{\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n}{2n}$ cosets for $D_n$ in $S_n$. This will also be the number of elements in a transversal for the left cosets of $D_n$ in $S_n$. Let the transversal be $t_1, \\ldots, t_l$. So, $S_n = t_1 D_n \\sqcup \\ldots \\sqcup t_l D_n$.  $\\text{\\bf 1}_{D_n} \\uparrow^{S_n}_{D_n}$ can be computed following Definition ~\\ref{def:induction}. \n\nIt would be instructive to discuss the character table of $S_n$ and $D_n$ here.\n\nComputing the character table of $S_n$ starts from computing the partitions $\\lambda_1 \\ge \\lambda_2 \\ge \\ldots \\ge \\lambda_r$ given $\\sum_i \\lambda = n$. These partitions can be partially ordered as follows. If there are two partitions $\\lambda = \\left(\\lambda_1 \\ge \\lambda_2 \\ge \\ldots\\right)$ and $\\mu = \\left(\\mu_1 \\ge \\mu_2 \\ge \\ldots\\right)$, $\\lambda \\ge \\mu$ if,\n\\begin{align}\n\\lambda_1 &\\ge \\mu_1\n\\nonumber\\\\\n\\lambda_1 + \\lambda_2 &\\ge \\mu_1 + \\mu_2\n\\nonumber\\\\\n\\lambda_1 + \\lambda_2 + \\lambda_3 &\\ge \\mu_1 + \\mu_2 + \\mu_3\n\\nonumber\\\\\n&\\vdots\n\\end{align}\n\nThe columns of the character table are indexed by the conjugacy classes such that the partitions are arranged in increasing order. On the other hand, the rows are indexed by the characters in the decreasing order of the partitions. Each cell in the table then contains the corresponding character.\n\nThe Lemma ~\\ref{lem:frob-recip-special} can be applied to obtain the Frobenius reciprocity for $D_n < S_n$.\n\n\\begin{align}\n\\langle \\chi_{\\text{\\bf 1}_{D_n}}, \\chi_\\rho \\downarrow^{S_n}_{D_n} \\rangle_{D_n} &= \\langle {\\chi \\uparrow^{S_n}_{D_n}}_{\\text{\\bf 1}_{D_n}} , \\chi_\\rho \\rangle_{S_n}\n\\end{align}\nwhere the left inner product is calculated in $S_n$ and the right one in $D_n$. So, if  $\\langle \\chi_{\\text{\\bf 1}_{D_n}}, \\chi_\\rho \\downarrow^{S_n}_{D_n} \\rangle_{D_n}$ needs to be determined, it would be sufficient to determine $\\langle {\\chi \\uparrow^{S_n}_{D_n}}_{\\text{\\bf 1}_{D_n}} , \\chi_\\rho \\rangle_{S_n}$.\n\nFollowing the Definition ~\\ref{def:inner-prod-char}, \n\n\\begin{align}\n\\langle {\\chi \\uparrow^{S_n}_{D_n}}_{\\mathbf{ 1}_{D_n}} , \\chi_\\rho \\rangle_{S_n} &= \\sum_{g_i \\in S_n}  {\\chi \\uparrow^{S_n}_{D_n}}_{\\mathbf{ 1}_{D_n}} (g_i) \\chi^\\dagger_\\rho (g_i)  \n\\nonumber\\\\\n&=\\sum_{g_i \\in S_n} \\left(\\sum^n_{i=1} \\delta_{{{ \\chi \\uparrow^{S_n}_{D_n}}_{\\mathbf{ 1}_{D_n}} (g_i)}_{i,i}}\\right)   \\chi^\\dagger_\\rho (g_i)  \n\\end{align}\n\n\nThe Theorem ~\\ref{theo:frob-char} is applied to determine the probabilities of measuring the irreducible representations of $D_n$ through proving a few lemmas. \n\n\\begin{mylem}\n\\label{lem:d-n-1-d-rep-prob-zero}\nThe probability of measuring the labels of one dimensional irreducible representations of $D_n$ is zero for non-trivial representations.\n\\end{mylem}\n\n\\begin{proof}[Proof of Lemma ~\\ref{lem:d-n-1-d-rep-prob-zero}]\nFirst, it is assumed that $n$ is even. So, there are four one dimensional representations and $\\frac{n-2}{2}$ two dimensional representations. The probability of measuring the one dimensional representations is then computed. Let the representations be denoted as $\\rho_1$, $\\rho_2$, $\\rho_3$, and $\\rho_4$. So, their dimensions are all the same i.e. $d_{\\rho_1} = d_{\\rho_2} = d_{\\rho_3} = d_{\\rho_4} = 1$. Let the probability of measuring $\\rho_i$ be $p_{\\rho_i}$. So, following Theorem ~\\ref{theo:frob-char},\n\n\\begin{align}\np_{\\rho_i} &= \\frac{|D_n|}{|S_n|} d_{\\rho_i} \\langle \\chi_{\\rho_i}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\nonumber\\\\\n&= \\frac{2 n}{n!}  \\langle \\chi_{\\rho_i}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\end{align}\n\n$D_n$ has $2 n$ elements. So, \n\n\\begin{align}\n\\langle \\chi_{\\rho_i}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{|D_n|} \\sum_{g\\in D_n} \\chi_{\\rho_i} \\left(g\\right) \\chi^\\dagger_{\\text{\\bf 1}_{D_n}} \\left(g\\right)\n\\nonumber\\\\\n&= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_i} \\left(g\\right) \n\\end{align}\n\nWhen $i = 1$, $\\rho_1$ is the trivial representation which sends all group elements to the $1 \\times 1$ matrix $\\begin{pmatrix}1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_1}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_1} \\left(g\\right) \n\\nonumber\\\\\n&= 1\n\\end{align}\n\nSo, \n\\begin{align}\np_{\\rho_1} &=  \\frac{2 n}{n!}  \\langle \\chi_{\\rho_1}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\nonumber\\\\\n&=  \\frac{2 n}{n!}  \n\\end{align}\n\nWhen $i = 2$, $\\rho_2$ is the representation which sends all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_2}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_2} \\left(g\\right) \n\\end{align}\n\nAs observed from the presentation of $D_n$, the number of elements in $\\langle x \\rangle$ is $n$. So, $n$ elements will be mapped to $1$ and $n$ elements will e mapped to $-1$. So,\n\n\\begin{align}\n\\langle \\chi_{\\rho_2}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\left(n \\left(1\\right) + n \\left(-1\\right)\\right) \n\\nonumber\\\\\n&= 0\n\\end{align}\n\nSo, \n\\begin{align}\np_{\\rho_2} &=  \\frac{2 n}{n!}  \\langle \\chi_{\\rho_2}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n}\n\\nonumber\\\\\n&= 0\n\\end{align}\n\nSo, the weak Fourier sampling will not be able to determine the labels of the sign representation which sends all elements in $\\langle x \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\nWhen $i = 3$, $\\rho_3$ is the representation which sends all elements in $\\langle x^2, y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_3}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_3} \\left(g\\right) \n\\end{align}\n\nFollowing Remark ~\\ref{rem:d-n-subgroup-order}, $\\rho_3$ sends $n$ elements of $D_n$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $2 n - n = n$ elements of $D_n$ to $\\begin{pmatrix}-1\\end{pmatrix}$. So,\n\n\\begin{align}\n\\langle \\chi_{\\rho_3}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= 0\n\\end{align}\n\nSo, the weak Fourier sampling will not be able to determine the labels of the sign representation which sends all elements in $\\langle x^2, y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\nWhen $i = 4$, $\\rho_4$ is the representation which sends all elements in $\\langle x^2, x y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $x$ to $\\begin{pmatrix}-1\\end{pmatrix}$. The probability of measuring this representation is given below.\n\n\\begin{align}\n\\langle \\chi_{\\rho_4}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\rho_4} \\left(g\\right) \n\\end{align}\n\nFollowing Remark ~\\ref{rem:d-n-subgroup-order}, $\\rho_4$ sends $n$ elements of $D_n$ to $\\begin{pmatrix}1\\end{pmatrix}$ and $2 n - n = n$ elements of $D_n$ to $\\begin{pmatrix}-1\\end{pmatrix}$. So,\n\n\\begin{align}\n\\langle \\chi_{\\rho_4}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= 0\n\\end{align}\n\nSo, the weak Fourier sampling will not be able to determine the labels of the sign representation which sends all elements in $\\langle x^2, x y \\rangle$ to $\\begin{pmatrix}1\\end{pmatrix}$ and all elements outside $\\langle x \\rangle$ to $\\begin{pmatrix}-1\\end{pmatrix}$.\n\nThe case when $n$ is odd may be considered as a special case of when $n$ is even and show that only the trivial representation can be sampled with non-zero probability.\n\\end{proof}\n\n\n\nNow, the probability of measuring the labels of two dimensional irreducible representations will be computed in weak Fourier sampling. The discussion starts with the following lemma.\n\n\\begin{mylem}\n\\label{lem:d-n-2-d-rep-prob-zero}\nThe probability of measuring the labels of two dimensional irreducible representations of $D_n$ is always zero.\n\\end{mylem}\n\n\\begin{proof}[Proof of Lemma ~\\ref{lem:d-n-2-d-rep-prob-zero}]\nWhen $n$ is even, there are $\\frac{n-2}{2}$ such irreducible representations. Let the irreducible representations be denoted as $\\sigma_1, \\sigma_2, \\ldots, \\sigma_k, \\dots, \\sigma_{\\frac{n-2}{2}}$.\n\nThe probability of measuring $\\sigma_k$ is given below.\n\n\\begin{align}\n\\langle \\chi_{\\sigma_k}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum_{g\\in D_n} \\chi_{\\sigma_k} \\left(g\\right) \n\\end{align}\n\nFollowing Remark ~\\ref{rem:char-dn}, $\\sigma_k$ maps $n$ number of elements to the matrices for which the characters of the representations are zero. For the rest $n$ number  of the group elements, the character is $2 \\cos \\left(\\frac{2 \\pi k l}{n}\\right)$ where $0\\le l \\le n-1$. So,\n\n\\begin{align}\n\\langle \\chi_{\\sigma_k}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{2 n} \\sum^{n-1}_{l = 0} 2 \\cos \\left(\\frac{2 \\pi k l}{n}\\right)\n\\nonumber\\\\\n&= \\frac{1}{ n} \\sum^{n-1}_{l = 0}  \\cos \\left(\\frac{2 \\pi k l}{n}\\right)\n\\end{align}\n\nThe following formula for the sum of series of cosines when they are in arithmetic progression is worth mentioning as they have been proven in both \\cite{knapp2009sines} and ~\\cite{holdener2009math}.\n\n\\begin{align}\n\\sum_{l=1}^{n} \\cos (l\\theta)=\\frac{\\sin(n\\theta\/2)}{\\sin(\\theta\/2)}\\cos ((n+1)\\theta\/2),\\quad \\sin(\\theta\/2)\\neq0.\n\\end{align}\n\nWhen $\\theta=\\dfrac{2 \\pi k}{n}$,\n\n\\begin{align}\n\\sum^n_{l=1} \\cos \\left(\\frac{2 \\pi k l}{n}\\right) &=\\frac{\\sin(\\pi k)}{\\sin(\\pi k\/n)}\\cos ((n+1)\\pi k\/n)= 0\n\\end{align}\n\nIf the interval of $l$ is changed from $\\left[1, n\\right]$ to $\\left[0, n-1\\right]$ the sum still remains zero.\n\n\\begin{align}\n\\sum^{n-1}_{l=0} \\cos \\left(\\frac{2 \\pi k l}{n}\\right) &= \\cos (0)+0-\\cos(2\\pi k)=0.\n\\end{align}\n\nSo, the probability of measuring the labels of the two dimensional irreducible representations is:\n\n\\begin{align}\n\\langle \\chi_{\\sigma_k}, \\chi_{\\text{\\bf 1}_{D_n}} \\rangle_{D_n} &= \\frac{1}{ n} \\sum^{n-1}_{l = 0}  \\cos \\left(\\frac{2 \\pi k l}{n}\\right)\n\\nonumber\\\\\n&= 0\n\\end{align}\n\nThe case of $n$ being odd can be considered as a special case of $n$ being even and the same result can be proved.\n\\end{proof}\n\nSo, the weak Fourier sampling algorithm cannot determine the labels of any of the two dimensional irreducible representations. To summarize, weak Fourier sampling cannot determine any irreducible representation other than the trivial one. So, it cannot determine the automorphism group of the cycle graph which is a trivial problem in the classical paradigm. Lemmas ~\\ref{lem:d-n-1-d-rep-prob-zero}  and ~\\ref{lem:d-n-2-d-rep-prob-zero} may be consolidated into the following theorem.\n\n\\begin{mytheo}\n\\label{theo:cycle-graph-auto-weak-fails}\nWeak quantum Fourier sampling fails to solve the cycle graph automorphism problem.\n\\end{mytheo}\n\n\\begin{proof}\nFollows directly from the proofs of the Lemmas ~\\ref{lem:d-n-1-d-rep-prob-zero}  and ~\\ref{lem:d-n-2-d-rep-prob-zero}.\n\\end{proof}\n\nIt is observed that the success probability of strong Fourier sampling is a conditional probability which depends on the success probability of measuring the labels of representation. As the success probability of measuring the non-trivial representations of $D_n$ is zero, the success probability of measuring their individual matrix elements is also zero.\n\n\\begin{mycor}\n\\label{cor:cycle-graph-auto-strong-fails}\nStrong quantum Fourier sampling fails to solve the cycle graph automorphism problem.\n\\end{mycor}\n\nSo far, the discussion in this paper has been limited to the POVMs in the computational basis. One could ask if the results are comparable if one  were performing entangled measurements introduced in \\cite{hallgren2010limitations, moore2005tight}. It can be argued that quantum Fourier transform is guaranteed to fail also in the latter type of measurement. To perform an entangled measurement, it is assumed that one has already measured the labels of the irreducible representations with nonzero probability which is not possible for cycle graphs. Hence, the automorphism group of a cycle graph cannot be computed using the measurements in either computational or entangled bases. It is also natural to ask how the results relate to the results of \\cite{radhakrishnan2005power} on using random bases. As the bases are always being chosen from the complete set of computational bases, one can argue that strong random Fourier sampling does not change the probability of measuring the labels of irreducible representations from zero (where applicable) to a larger value. It can also be argued in a similar manner for both Kuperberg sieve \\cite{kuperberg2005subexponential, moore2010impossibility} and Pretty Good Measurement (PGM) as both algorithms are conditioned on weak sampling first. The Ettinger-H{\\o}yer-Knill theorem \\cite{ettinger2004quantum} that the quantum query complexity of the hidden subgroup problem is polynomial may also be mentioned here. The theorem cannot be applied here either as the assumption of the theorem is that the quantum algorithm will always output a subset of the hidden subgroup.\n\n\\section{Quantum Fourier transform for other graph automorphism problems}\n\\label{sec:qft-ga-plus}\nIn Section ~\\ref{sec:weak-cycle}, it has been shown that the  Fourier sampling fails to determine the automorphism group of the cycle graphs. It is natural to ask whether same is the case for the graphs which have the dihedral group as a subgroup in it's automorphism group. The question can be answered in the affirmative. Those cases can be instantiated using any of the following three approaches.\n\n\\subsection{Inductive approach}\nThis approach starts with a graph whose automorphism group is a dihedral group. Then, more complicated structures are inductively built on that graph. This is based of the following theorem reproduced from \\cite{dresselhaus2007group} and also mentioned as Theorem $10$ in Section $3.2$ in \\cite{serre2012linear}. To initiate the discussion,  an example can be provided, where a more complex graph is constructed from cycle graphs and show that quantum Fourier sampling is still guaranteed to fail to compute the automorphism group.\n\n\\begin{myexamp}[QFT for the automorphism group of $C_m \\sqcup C_n$]\n\\label{examp:c-m-c-n}\nThe graph $C_m \\sqcup C_n$ can be visualized as follows.\n\nIt is already known from \\cite{ganesan2012automorphism, jordan1869assemblages} that the automorphism group of $C_m \\sqcup C_n$ is $D_m \\times D_n$ where the dihedral groups are of order $2 m$ and $2 n$ respectively.\n\nFollowing the schemes introduced in Section ~\\ref{sec:weak-cycle},  the probabilities of measuring the labels irreducible representations of $D_m \\times D_n$ are computed.\n\nThe Theorem $10 (ii)$ in \\cite{serre2012linear} indicates that every irreducible representation of $D_m \\times D_n$ can be determined from the irreducible representations of $D_m$ and $D_n$.\n\nThe general expression for irreducible representations of dihedral groups is used as it is  given in Section ~\\ref{sec:weak-cycle}. Let the $i$-th irreducible $j$-dimensional representation of a dihedral group $D_n$ be $\\rho_{i,j, n}$ where $n$ is even. For example, for $D_m$, the irreducible representations are $\\rho_{1,1, m}$, $\\rho_{2,1, m}$, $\\rho_{3,1, m}$, $\\rho_{4,1, m}$, $\\rho_{1,2, m}$, $\\rho_{2,2, m}$, $\\rho_{3,2, m}$, and $\\rho_{4,2, m}$. The expressions of these representations can be derived from Section ~\\ref{sec:irrepdn}. It needs to be noted that in the two dimensional irreducible representations, there is an additional ordering parameter $k$. Hence, $k_m$ and $k_n$ will be used for the groups $D_m$ and $D_n$ respectively. So, for example, $\\begin{pmatrix}\ne^{\\frac{2 \\pi i k_{n}}{n}}&0\\\\\n0&e^{-\\frac{2 \\pi i k_{n}}{n}}\n\\end{pmatrix}$ denotes the $k_{n}$-th two dimensional irreducible representation for the element $x$. Another parameter $l$ is used to identify the elements $x^l$ in the group $D_n$ where $0 \\le l \\le n-1$. Let the parameter be $l_m$ for $D_m$ and $l_n$ for $D_n$.\n\nNow, the irreducible representations of $D_m \\times D_n$ are enumerated. The corresponding characters are also computed at the same time. There are sixteen one dimensional, thirty two two dimensional, and sixteen four dimensional irreducible representations. Table ~\\ref{table:irrep-cn-cm} summarizes the characters of the irreducible representations of $D_m \\times D_n$.  Curious readers may refer to Appendix ~\\ref{app:cmcn} for detailed derivations.\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{ c| c   } \n \\hline\nIrreducible representation  & Character\\\\ \n \\hline\\hline\n  $\\rho_{7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47-64} $ &  $\\chi_{7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47-64} = 0$\\\\\n   $\\rho_{1}$ & $\\chi_{1} = 1$ \\\\\n $\\rho_{2-4, 9-12, 17-20, 25-28}$ & $\\chi_{2-4, 9-12, 17-20, 25-28} = \\pm 1$ \\\\\n $\\rho_5 $ &  $\\chi_5 =  2 \\cos \\left(\\frac{2 \\pi k_m}{m}\\right)$\\\\\n $\\rho_{6, 22} $ &  $\\chi_{6, 22} = 2 \\cos \\left(\\frac{2 \\pi k_m l_m}{m}\\right)$\\\\\n $\\rho_{13, 21, 29} $ & $\\chi_{13, 21, 29} = \\pm 2 \\cos \\left(\\frac{2 \\pi k_m}{m}\\right)$\\\\\n $\\rho_{14, 30} $ & $\\chi_{14, 30} = \\pm  2 \\cos \\left(\\frac{2 \\pi k_m l_m}{m}\\right)$\\\\\n $\\rho_{33} $ & $\\chi_{33} = 2 \\cos \\left(\\frac{2 \\pi k_n}{n}\\right)$\\\\\n $\\rho_{34-36}$ & $\\chi_{34-36} =\\pm 2 \\cos \\left(\\frac{2 \\pi k_n}{n}\\right)$\\\\\n $\\rho_{37} $ & $\\chi_{37} = 4 \\cos \\left(\\frac{2 \\pi  k_m}{m}\\right) \\cos \\left(\\frac{2 \\pi  k_n}{n}\\right)$\\\\\n $\\rho_{38} $ & $\\chi_{38} = 4 \\cos \\left(\\frac{2 \\pi  k_n}{n}\\right) \\cos \\left(\\frac{2 \\pi  k_m l_m}{m}\\right)$\\\\\n $\\rho_{41} $ & $\\chi_{41} = 2 \\cos \\left(\\frac{2 \\pi k_n l_n}{n}\\right)$\\\\\n $\\rho_{42-44} $ & $\\chi_{42-44} =\\pm 2 \\cos \\left(\\frac{2 \\pi k_n l_n}{n}\\right)$\\\\\n $\\rho_{45}$ & $\\chi_{45} = 4 \\cos \\left(\\frac{2 \\pi  k_m}{m}\\right) \\cos \\left(\\frac{2 \\pi  k_n l_n }{n}\\right)$\\\\\n $\\rho_{46} $ & $\\chi_{46} = 4 \\cos \\left(\\frac{2 \\pi  k_m l_m}{m}\\right) \\cos \\left(\\frac{2 \\pi  k_n l_n }{n}\\right)$\\\\\n\\hline\n\\end{tabular}\n\\caption{Irreducible representations of $\\text{Aut} \\left(C_m \\sqcup C_n\\right)$}\n\\label{table:irrep-cn-cm}\n\\end{table}\n\nIt can be readily seen that the quantum Fourier transform is guaranteed to fail to measure the labels of   $\\rho_{7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47-64}$. With little more algebraic steps, it can be shown to be true for few more irreducible representations.\n\\end{myexamp}\n\nFollowing Example ~\\ref{examp:c-m-c-n}, one can create arbitrarily large classes of graph automorphism problem by based on the known Theorem ~\\ref{lab:irrep-prod-grp} (not a result original to this paper) for which quantum Fourier transform will always fail. The proof of the Theorem ~\\ref{lab:irrep-prod-grp} is given as Theorem $10$ in \\cite{serre2012linear}\n\n\\begin{mytheo}\n\\label{lab:irrep-prod-grp}\nThe direct product of two irreducible representations of groups $H$ and $K$ yields an irreducible representation of the direct product group so that all irreducible representations of the direct product group can be generated from the irreducible representations of the original groups before they are joined.\n\\end{mytheo}\n\nThis paper provides Algorithm ~\\ref{algo:cycle-graph-infinite} which is a way to generate arbitrarily large class of graph automorphism problems for which quantum Fourier sampling is guaranteed to fail.\n\n\\begin{algorithm}[H]\n\\label{algo:arb-fail}\n\\caption{An algorithm to create arbitrarily large easy graph automorphism problem}\n     \\label{algo:cycle-graph-infinite}\n\\begin{algorithmic}[1]\n\\Procedure {DISJOINT-CYCLE-GRAPH}{$m, n$} \\Comment{$m$ is the number of nodes for the smallest cycle, $n$ is the number of cycles}\n\n\\State Create an $m$-cycle graph \\Comment{The first cycle}\n\n\\For{$i\\gets 1, n-1 $}\n\\State Increase $m$ by one\n\\State Create an $m$-cycle graph \\Comment{The next cycle}\n\\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\nBy the end of its execution, Algorithm ~\\ref{algo:cycle-graph-infinite} will generate a graph of $n$ cycles with a total of $m n$ nodes and the time complexity will be $O\\left(poly(m, n)\\right)$. One can be more creative about the Step 4 of Algorithm ~\\ref{algo:cycle-graph-infinite} to create other classes of graph automorphism problems for which quantum Fourier sampling is guaranteed to fail.\n\nAt this point following remark can be made.\n\n\\begin{myrem}\n\\label{rem:dn-prod-g-qft-fail}\nArbitrarily large classes of graph automorphism problems can be created for which quantum Fourier sampling is guaranteed to fail. Quantum Fourier sampling is guaranteed to fail to compute the automorphism group of a graph when the automorphism group is the product of a dihedral group and any finite group.\n\\end{myrem}\n\n\\subsection{Existential approach}\nIn the existential approach, a general class of graphs is chosen and it is proven that there is at least one graph in that class for which quantum hidden subgroup algorithm is guaranteed to fail to compute the automorphism group.\n\nThe discussion starts with the Frucht's theorem \\cite{frucht1939herstellung}.\n\n\\begin{mytheo}\nEvery abstract group is isomorphic to the automorphism group of some graph.\n\\end{mytheo}\n\nSo, any group which is a product of $D_n$ and a finite group $G$ is isomorphic to the automorphism group of some finite graph. It has also been shown in \\cite{babai1996automorphism} that  every finite group as the group of symmetries of a strongly regular graph. It indicates that there is a class of strongly regular graph whose automorphism group is isomorphic to $D_n \\times G$. According to the Theorem ~\\ref{lab:irrep-prod-grp} and Remark ~\\ref{rem:dn-prod-g-qft-fail}, one can argue that  quantum Fourier sampling should fail to construct the automorphism group of a subclass of strongly regular graphs.\n\nAnother example may be the  Cayley graph automorphism problem \\cite{xu1998automorphism}. It is well known that the automorphism group of the Cayley graph $\\text{Aut}\\left( C \\left(G, X\\right)\\right)$ of a group $G$ over a generating set $X$ contain an isomorphic copy of $G$ acting via left translations \\cite{jajcay2000structure}. In that case, the automorphism group of the Cayley graph of a dihedral group $D_n$  contains $D_n$ as a subgroup. So, following the result of the previous section, quantum Fourier sampling fails to compute the automorphism group of $\\text{Aut}\\left( C \\left(G, X\\right)\\right)$.\n\n\\subsection{Universal structures approach}\nA class $\\mathcal{C}$ of structures  is called {\\it universal} if every finite group is the automorphism group of a structure in $\\mathcal{C}$  \\cite{cameron2004automorphisms}. A series of works by Frucht, Sabidussi, Mendelsohn, Babai, Kantor, and others \\cite{cameron2004automorphisms} has shown the following classes of graphs to be universal - graphs of valency $k$ for any fixed $k > 2$ \\cite{frucht1949graphs}; bipartite graphs; strongly regular graphs \\cite{mendelsohn1978every}; Hamiltonian graphs \\cite{sabidussi1957graphs}; $k$-connected graphs \\cite{sabidussi1957graphs}, for $k > 0$; $k$-chromatic graphs, for $k > 1$; switching classes of graphs; lattices \\cite{birkhoff1946grupos}; projective planes (possibly infinite); and Steiner triple systems \\cite{mendelsohn1978groups}; and symmetric designs (BIBDs). It indicates that each of these classes has at least one graph which has its automorphism group isomorphic to $D_n \\times G$ where $G$ is any finite group. So, quantum Fourier sampling will fail to compute the automorphism group of each of these cases.\n\n\\section{Is hidden subgroup the ideal approach?}\nIt has been shown that there are instances of graph automorphism problem for which hidden subgroup algorithm can never be successful although they are trivial to solve on a classical computer. So, it can be argued that the space of the hidden subgroup representations of all graph automorphism problems cannot capture the structure of the space of all graph automorphism problem. As the graph isomorphism problem is believed to be at least as hard as the graph automorphism problem, it can also be added that the space of the hidden subgroup representations of all graph isomorphism problem cannot capture the structure of the space of all graph isomorphism problem. So, it would be appropriate to investigate alternative quantum algorithmic approach for these classes of problems.\n\n\\section{Conclusion}\nIt has been shown that, while solving the hidden subgroup representation of the graph isomorphism problem is equivalent to determining order $2$ subgroup of a symmetric group,  the hidden subgroup representation of the graph automorphism problem is equivalent to determining a hidden subgroup of higher order. This paper has identified a class of graph automorphism problem for which the quantum Fourier transform algorithm always fails. It also has shown how one can determine non-trivial classes of graphs for which the same algorithm always fails. With these negative results, one may be interested to ask whether the hidden subgroup representation is a practical representation of the graph isomorphism and automorphism problems in quantum regime.\n\n\\section*{Acknowledgement}\nOS thanks Dave Bacon, Aram Harrow, Robert Campbell, Marc Bogaerts, Andrew Childs,  Steven Gregory, Jef Laga, Dietrich Burde, Eric Wofsey, Alexander Hulpke, Michael Burr, Jyrki Lahtonen, Joshua Grochow and  Tobias Kildetoft for their helpful comments.\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nMassive black holes (BHs) with masses >$10^{6}$$M_{\\odot}$\\ reside in most galaxies, including our own \\citep{Genzel:1996aa, Magorrian:1998aa, Schodel:2003aa,Ghez:1998aa, Ghez:2000aa, Ghez:2008aa}. Luminous quasars with $M_\\mathrm{BH}$\\ $\\sim10^{9}$$M_{\\odot}$\\ (\\citealt{Barth:2003ab}: $z = 6.4$, \\citealt{Willott:2003aa}: $z = 6.41$, \\citealt{Trakhtenbrot:2011aa}: $z\\sim4.8$) have been detected at $z\\sim5-7$ (\\citealt{Fan:2000aa, Fan:2001aa}: $z\\sim6$, \\citealt{Mortlock:2011aa}: $z = 7.085$). The BHs powering these quasars must therefore build up their mass in less than one billion years. Depending on the assumed seed formation model, almost constant Eddington accretion or even super-Eddington episodes are required to match these observations \\citep{Volonteri:2005aa, Volonteri:2014aa, Alexander:2014aa}.\nIn our current understanding, BHs grew out of $\\sim100-10^{5}$$M_{\\odot}$\\ seeds by accreting infalling matter or merging with a second BH \\citep{Rees:2007aa}.  \nTwo seed formation models are currently favored. One scenario predicts that the remnants of massive Population III (Pop III) stars constitute BH progenitors \\citep{Madau:2001aa, Haiman:2001aa, Bromm:2002aa, Alvarez:2009aa, Johnson:2012aa}. The second model is based upon the direct gravitational collapse of massive gas clouds \\citep{Loeb:1994aa, Bromm:2009aa, Volonteri:2010aa, Latif:2013ab}. Both models include uncertainties and predict markedly different BH growth histories. More exotic scenarios, including BH seed formation via stellar dynamical, rather than gas dynamical processes, have also been suggested ( see \\citealt{Volonteri:2010aa, Bromm:2011aa} and references therein). These scenarios primarily focus on reproducing the high-redshift quasar population. We must however also be able to explain the existence of less massive and luminous, but more abundant BHs that we find in galaxies such as the Milky Way \\citep{Treister:2011aa, Volonteri:2010aa}. A first step towards constraining seed formation models and determining if they are also valid for such 'normal' BHs, is measuring the BH luminosity function at high redshift.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.49\\textwidth]{goods_scat_plot}\n\\end{centering}\n\\caption{\\label{fig:offset}Offset between GOODS\/ACS and \\textit{Chandra}\\ positions after the correction has been applied. Out of the 740 \\textit{Chandra}\\ X-ray sources 408 possess an optical counterpart. We determine a mean offset of (0.128$^{\\prime\\prime}$, -0.237$^{\\prime\\prime}$) for these 408 objects and correct for it by shifting the \\textit{Chandra}\\ positions. The grey triangle indicates the mean displacement before the correction. The green star illustrates the mean offset after our correction. The black points show the corrected object positions. The grey circle illustrates an offset of 0.5$^{\\prime\\prime}$.}\n\\end{figure}\n\nThe \\textit{Chandra}\\ 4-Ms catalog \\citep{Xue:2011aa} provides X-ray counts for the soft (0.5 keV - 2 keV), hard (2 keV - 8keV) and full (0.5 keV - 8 keV) band for the \\textit{Chandra}\\ Deep Field South (CDF-S). The on-axis flux limits lie at $9.1 \\times 10^{-18}$, $5.5 \\times 10^{-17}$ and $3.2 \\times 10^{-17} \\mathrm{erg}\\ \\mathrm{s}^{-1}\\ \\mathrm{cm}^{-2}$ for the soft, hard and full band, respectively. The CDF-S covers a $0.11$ $\\mathrm{deg}^2$ area. For our analysis we not only use the \\textit{Chandra}\\ 4-Ms data, but also require coverage by the CANDELS wide and deep surveys. The effective area of our field is hence 0.03 $\\mathrm{deg}^2$ \\footnote{README CANDELS GOODS-S Data Release v1.0 \\url{http:\/\/archive.stsci.edu\/pub\/hlsp\/candels\/goods-s\/gs-tot\/v1.0\/hlsp_candels_hst_acs-wfc3_gs-tot_readme_v1.0.pdf}}. \n\n\\cite{T13} used the \\textit{Chandra}\\ 4-Ms data in their search for high-redshift ($z > 6$) AGN. Using a sample of preselected $z = 6 - 8$ Lyman Break dropout and photometrically selected sources from the HUDF and CANDELS, \\cite{T13} showed that none of these sources are detected individually in the X-rays. Stacking the X-ray observations does not produce a significant detection either. \\cite{T13} suggested different processes that could account for the lack of X-ray counterparts to these high-redshift sources. \nThe sample could be contaminated by a large number of low-redshift interlopers. \nA low BH occupation fraction could explain the lack of X-ray counterparts. \nIt is possible that BH growth only occurs in dusty and\/or small galaxies which were not included in this analysis because they lie below the detection threshold. \nThe Treister et al. results can also be explained if large amounts of gas and dust obscure the X-ray emission of actively accreting BHs, \nas previously proposed by \\cite{Treister:2009aa} and \\cite{Fiore:2009aa}.  \nAdditionally, it is possible that accretion is not the dominant BH growth mode in the early universe. If BHs are primarily gaining mass by merging with other BHs, X-ray radiation might not probe BH activity. \n(See \\citealt{T13} for a more detailed discussion of these possible scenarios.)\nIn addition to the scenarios described above, short, super-Eddington growth episodes, as proposed by \\cite{Madau:2014aa} and \\cite{Volonteri:2014aa}, also present a possible solution. \nIn comparison to constant Eddington accretion, the amount of matter that is accreted during these super-Eddington growth phases is the same. However, \\cite{Madau:2014aa} showed that, if we allow super-Eddington accretion, a duty cycle of $20\\%$ is enough to grow a non-rotating $100$ $M_{\\odot}$\\ seed BH into a $10^9$ $M_{\\odot}$\\ object by $z\\sim7$. One could imagine that the seed BH grows via five $20\\ \\mathrm{Myr}$ long $\\dot{m}\/\\dot{m}_\\mathrm{Edd}=4$ growth modes, each followed by a $100\\ \\mathrm{Myr}$ phase of quiescence. For short, super-Eddington growth episodes, we would thus expect to find fewer BHs that are actively accreting at the same time. The \\cite{T13} sample could therefore not contain any BHs that are actively accreting at the time of observation. \n\n\\label{sec:AnalysisSteps}\n\\begin{figure*}\n\\begin{centering}\n\\input{MindMap2.tex}\n\\caption{\\label{fig:flowchart} Flowchart illustrating the analysis steps that we take to determine possible $z\\gtrsim5$ candidates. The number of objects that pass each step is given in brackets. After executing this analysis for all 740 Chandra sources, three $z\\gtrsim5$ candidates remain in our sample. Additional $K_S$-band and deep $Y$-band data does however show that they are most likely low-redshift sources.}\n\\end{centering}\n\\end{figure*}\n\nIn this work we carefully examine the \\textit{Chandra}\\ 4-Ms catalog for possible $z\\gtrsim5$ AGN. We combine the deep \\textit{Chandra}\\ observations with optical and infrared data from GOODS, CANDELS and \\textit{Spitzer}. We use the Lyman Break Technique and a photometric redshift code to estimate the redshift of our targets. We also use colour criteria, stacking and the X-ray Hardness Ratio. In contrast to \\citealt{T13}, we base this analysis on detected X-ray sources instead of trying to determine if a high-redshift object possesses a X-ray counterpart. We therefore also analyse X-ray sources that would not be classified as Lyman Break Galaxies because they are heavily obscured in the optical and the infrared. \n\nWe know that the CDF-S contains hundreds of well constrained $z\\gtrsim5$ Lyman Break Galaxies (see e.g. \\citealt{Stark:2009aa, Vanzella:2009aa, Wilkins:2010aa, Bouwens:2014aa, Duncan:2014aa}). These should all pass our manual inspection, stacking, colour criteria and photometric redshift measurement. To be considered as a high-redshift AGN candidate, they must however also be detected in the X-rays and pass our X-ray Hardness Ratio test.\n\n\\cite{Volonteri:2010ab} showed that the expected number density of high-redshift AGN depends on the assumed seed formation model. Both, a detection and a non-detection, of high-redshift AGN gives us a lower limit on this number density. Our search thus constrains possible seed formation scenarios and sheds light on BH growth modes.  \n\n\\cite{Vito:2013aa} searched for $z > 3$ AGN in the 4-Ms CDF-S. They mainly analysed the evolution of obscuration and AGN space density with redshift. In contrast to this work, \\cite{Vito:2013aa} based their analysis on already existing photometric and spectroscopic information on the \\textit{Chandra}\\ sources. We compare our results to \\cite{Vito:2013aa} in Section \\ref{sec:discussion}. \n\nThis paper is organized as follows. Section \\ref{sec:data} describes the data that is used in this work. Sections \\ref{sec:analysis} and \\ref{sec:combination} introduce our redshift tests and illustrate the results of their combination. We conclude with a discussion in Section \\ref{sec:discussion} and a summary in section \\ref{sec:summary}. Throughout this paper we assume a $\\Lambda$CDM cosmology with $h_0$ = 0.7, $\\Omega_\\mathrm{m}$ = 0.3 and $\\Omega_\\Lambda$ = 0.7. All magnitudes are given in the AB system \\citep{Oke:1983aa}.\n\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.49\\textwidth]{CDFs_goods_candels_field}\n\\end{centering}\n\\caption{\\label{fig:fields}Overview of the area covered by the HST\/ACS filters $B$ (red), $V$, $i$, $z$ (blue), the HST\/WFC3 bands $Y$ (green), $J$, $H$ (purple) and our 740 \\textit{Chandra}\\ 4-Ms sources. Grey points mark sources that are not of interest for this work because they are not covered by enough bands ($B$, $V$, $i$, $z$, $J$, $H$) to use the Lyman Break Technique. Black points indicate objects with enough filter coverage that were eliminated because the objects are clearly visible in all bands. According to the Lyman Break Technique this indicates $z < 4$. Yellow stars mark the positions of the 58 potential high-redshift AGN that we analyse more closely.}\n\\end{figure}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{photometry_measuredSens}\n\\caption{\\label{fig:comp_phot} Comparison between our own photometry and the flux values reported in the GOODS catalog  \\citep{Giavalisco:2004aa}. We show the AB magnitude values for the $z$-band. Note that the GOODS\/ACS catalog only contains flux values for an aperture with a 0.707$^{\\prime\\prime}$ and not a 0.75$^{\\prime\\prime}$ radius. Nonetheless, we compare these values to the brightness we measured within a 0.75$^{\\prime\\prime}$ radius aperture. We use a small 0.3$^{\\prime\\prime}$ radius aperture for two objects only. Since these two sources are not detected in the GOODS\/ACS catalog, they are not part of this comparison.}\n\\end{centering}\n\\end{figure}\n\n\\section{Data}\n\\label{sec:data}\n\nThe \\textit{Chandra}\\ 4-Ms source catalog by \\cite{Xue:2011aa} is the starting point of this work. It contains 740 sources and provides counts and observed frame fluxes in the soft (0.5 keV - 2 keV), hard (2 keV - 8 keV) and full (0.5 keV - 8 keV) band. All object IDs used in this work refer to the source numbers listed in the \\cite{Xue:2011aa} \\textit{Chandra}\\ 4-Ms catalog.\nWe make use of HST\/ACS data from the GOODS-south survey in the optical wavelength range. We use catalogs and images for filters F435W ($B$), F606W ($V$), F775W ($i$) and 850LP ($z$) from the second GOODS\/ACS data release (v2.0) \\citep{Giavalisco:2004aa}. \nWe use CANDELS WFC3\/IR data from the first data release (v1.0) for passbands F105W ($Y$), F125W ($J$) and F160W ($H$) \\citep{Grogin:2011aa,Koekemoer:2011aa}. \nTo determine which objects are red, dusty, low-redshift interlopers, we also include the 3.6 micron and 4.5 micron \\textit{Spitzer}\\ IRAC channels. We use SIMPLE  image data from the first data release (DR1, \\citealt{van-Dokkum:2005aa}) and the first version of the extended SIMPLE catalog by \\cite{Damen:2011aa}.  \n\nWhen comparing \\textit{Chandra}, GOODS\/ACS and CANDELS object positions, a clear offset in the \\textit{Chandra}\\ coordinates is apparent. We illustrate this inconsistency in Figure \\ref{fig:offset}. To correct for this discrepancy, we calculate  the mean displacement between the \\textit{Chandra}\\ and the GOODS\/ACS catalog. We determine a mean offset of $\\mathrm{RA}_{Chandra}-\\mathrm{RA}_\\mathrm{ACS}=0.128^{\\prime\\prime}$ and $\\mathrm{DEC}_{Chandra}-\\mathrm{DEC}_\\mathrm{ACS}=-0.237^{\\prime\\prime}$.  We adjust the GOODS\/ACS and CANDELS positions of each object by subtracting the mean displacement from the originally given catalog position. \n \n\\section{Analysis}\n\\label{sec:analysis}\nIn the following section we describe in detail the set of criteria we employed to the data in order to identify $z\\gtrsim5$ candidates. We first exclude objects with insufficient filter coverage, perform our own aperture photometry and determine the dropout band of each source by manual inspection. We  run a photometric redshift code, stack the GOODS\/ACS data and apply colour criteria. In addition, we use the X-ray data as a photometric redshift indicator. We combine all redshift tests in section \\ref{sec:combination}. Figure \\ref{fig:flowchart} illustrates and summarizes the complete analysis that is detailed in the following subsections. \n\n\\cite{Dahlen:2010aa}, \\cite{McLure:2011aa} and \\cite{Duncan:2014aa} showed that when selecting high-redshift objects, a selection based only on colour criteria is not as reliable as calculating photometric redshifts. Especially for faint, low signal-to-noise objects, errors and upper limits can lead to scattering out of the colour selection region. Similar to colour criteria, photometric redshifts strongly depend on the position of the Lyman Break. A photometric redshift code does however consider all filter information, including upper limits and errors. Furthermore, low-redshift interlopers can be identified by including the filters redward of the Lyman Break. \\cite{Dahlen:2010aa} illustrated the discrepancy between colour criteria and photometric redshifts for the GOODS-S field. Only 50$\\%$ of their photometrically selected $z\\sim4$ sources were also classified as $B$-dropouts according to colour criteria. We therefore primarily use a photometric redshift code. The colour criteria, our visual classification, the stacking of the GOODS\/ACS data and the X-ray Hardness Ratio provide additional redshift indications. \n\n\\subsection{Initial Sample Selection}\nOur initial sample consists  of the 740 objects given in the \\textit{Chandra}\\ 4-Ms source catalog. Figure \\ref{fig:fields} shows that not all \\textit{Chandra}\\ targets are covered by the GOODS\/ACS and CANDELS images. However, adjacent filter coverage is necessary for the application of the Lyman Break Technique. We therefore narrow the number of possible candidates down to 374 by removing sources that are not covered by $B$, $V$, $i$, $z$, $J$ and $H$. The $Y$-band area is small compared to the other filters. We therefore also include sources that are not covered by the $Y$-band provided that they are covered by all other GOODS\/ACS and CANDELS filters. The optical and infrared counterpart detection is primarily based upon the $H$-band image since this is the deepest band. The $H$-band images for objects 105 and 521 show significant artifacts, we therefore discard them. Source 366 is eliminated due to the object's position being at the edge of the GOODS\/ACS images. Hence, 371 possible candidates remain after this first visual preselection.  \n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{208_series_stamps}\n\\end{minipage}\n\\begin{minipage}{\\textwidth}\n$\\rightarrow \\textit{eliminated: lack of CANDELS coverage}$\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{141_series_stamps}\n\\end{minipage}\n\\begin{minipage}{\\textwidth}\n$\\rightarrow \\textit{eliminated: source is clearly visible by eye in the optical and the infrared, this indicates $z < 4$ according to}$\\\\\n$\\textit{the Lyman Break Technique}$\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{184_series_stamps}\n\\end{minipage}\n\\begin{minipage}{\\textwidth}\n$\\rightarrow  \\textit{kept in sample: by eye classified as a $B$-dropout, this indicates $z\\sim4$ according to the Lyman Break Technique}$\n\\end{minipage}\n\n\\caption{\\label{fig:visclass}Classification examples. We only kept source 184 in our sample. The images are 10$^{\\prime\\prime}$ x 10$^{\\prime\\prime}$ in size and were colour inverted. }\n\\end{figure*}\n\n\n\\begin{figure*}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{190_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{280_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{333_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{384_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{643_series_stamps}\n\\end{minipage}\n\n\\begin{center}\n\\line(1,0){450}\n\\end{center}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{104_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{156_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{276_series_stamps}\n\\end{minipage}\n\n\\caption{\\label{fig:mysteryobjects}Sources that can not be classified according to their dropout band. $Top$: Sources that are in the \\textit{Chandra}\\ 4-Ms catalog but do not show a clear counterpart in the optical and infrared, \\textit{'low significance objects'}. 333, 384 and 643 are close to a bright galaxy which is why we might not be able to detect the counterpart in the optical and infrared. Deeper observations would be needed to detect possible counterparts. $Bottom$: Sources with multiple possible counterparts. Out of these eight objects only 156, 276 and 333 are simultaneously detected in the hard and in the soft band. We are unable to gain redshift estimates for these eight objects and are thus unable to determine if they are low or high-redshift sources or if they might be spurious detections. The black circles are centered on the original \\textit{Chandra}\\ position and illustrate the positional uncertainty given in the \\protect\\cite{Xue:2011aa} catalog. All images were colour inverted and are 10$^{\\prime\\prime}$ x 10$^{\\prime\\prime}$ in size.} \n\\end{figure*}\n\n\\subsection{Aperture Photometry}\n\\label{sec:aperturephot}\nWe perform our own aperture photometry on the GOODS\/ACS and CANDELS images to gain flux values and to estimate parameters such as detection threshold and aperture size. We compare our results to the GOODS\/ACS catalog (Figure \\ref{fig:comp_phot}). \n\nTo perform aperture photometry we use Source-Extractor (SExtractor, \\citealt{Bertin:1996aa, 2002Terapix, Holwerda:2005aa}). We determine the counterpart position in the $H$ band in a first SExtractor run. We then run SExtractor on the remaining optical and infrared images to establish if the counterpart is present at the same location. The flux measurements are carried out within circular apertures with radii between 0.3$^{\\prime\\prime}$ and 1$^{\\prime\\prime}$. We alter the aperture size for faint sources and to prevent contamination through nearby objects. For sources with a signal-to-noise ratio $<1$ we use the $1\\sigma$ sensitivity limit of the corresponding filter as an upper limit. The SExtractor parameter values and aperture sizes are summarized in Table \\ref{tab:SE_para} and Table \\ref{tab:SE_ID}. For the 3.6 micron and 4.5 micron bands we rely on the flux values reported in the SIMPLE catalog \\citep{Damen:2011aa}. We make use of the flux values reported for a 1.5$^{\\prime\\prime}$ radius aperture. We show flux and error values for our main sample in Tables \\ref{tab:apphot1} and \\ref{tab:apphot2}.\n\n\n\\subsection{Lyman Break Technique and visual classification}\n\\label{sec:LBT}\nThe Lyman Break Technique \\citep{Steidel:1999fj, Giavalisco:2002aa, Dunlop:2013zr} employs the pronounced feature of the Lyman continuum discontinuity in the spectral energy distribution (SED) of young, star forming galaxies. We use the common terminology of referring to Lyman Break Galaxies as 'dropouts'. If a source is not detected in the $B$-band or any bluer passbands, but is visible in all redder filters, this indicates $z\\sim4$ and we refer to it as a '$B$-dropout'. $V$-dropouts, $i$-dropouts and $z$-dropouts correspond to redshifts of $\\sim5$, $\\sim6$ and $\\sim7$ respectively. \n\nWe classify the 371 possible candidates by eye according to their dropout band. If an object is clearly visible in all bands, this indicates $z < 4$. We exclude such sources from our sample. Figure \\ref{fig:visclass} illustrates the conditions sources have to fulfill to be included in the further analysis. \\\\\n\nEight sources are not classified according to their dropout band. These objects are shown in Figures \\ref{fig:mysteryobjects}. Figure \\ref{fig:counts} shows the hard and soft band counts for all eight sources in comparison to the entire sample. The hard, soft and full band counts are given in Table \\ref{tab:mysobscounts}.\\\\ \n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\begin{tabular}{llllllllll}\n\t\t\\toprule\n\t\t\t{ID} & {Hard counts} & {$\\sigma_\\mathrm{Hard}$} & {$\\mathrm{SNR}_\\mathrm{Hard}$} & {Soft counts} & {$\\sigma_\\mathrm{Soft}$} & {$\\mathrm{SNR}_\\mathrm{Soft}$} & {Full counts} & {$\\sigma_\\mathrm{Full}$} & {$\\mathrm{SNR}_\\mathrm{Full}$}\\\\\n\n\t\t\t\\midrule\n\t\t\t$190$ & $27.55$ & $-1.00$ & $\\mathrm{-}$ & $17.29$ & $-1.00$ & $\\mathrm{-}$ & $21.99$ & $10.12$ & $2.17$\\\\\n\t\t\t$280$ & $15.13$ & $-1.00$ & $\\mathrm{-}$ & $9.36$ & $4.94$ & $1.89$ & $18.36$ & $-1.00$ & $\\mathrm{-}$\\\\\n\t\t\t$333$ & $51.15$ & $16.91$ & $3.02$ & $52.23$ & $11.59$ & $4.51$ & $103.21$ & $19.66$ & $5.25$\\\\\n\t\t\t$384$ & $11.37$ & $-1.00$ & $\\mathrm{-}$ & $7.25$ & $4.44$ & $1.63$ & $14.98$ & $-1.00$ & $\\mathrm{-}$\\\\\n\t\t\t$643$ & $34.34$ & $-1.00$ & $\\mathrm{-}$ & $27.16$ & $8.72$ & $3.11$ & $33.54$ & $13.21$ & $2.54$\\\\\n\t\t\t\\midrule\n\t\t\t$104$ & $49.95$ & $-1.00$ & $\\mathrm{-}$ & $38.57$ & $12.16$ & $3.17$ & $57.78$ & $-1.00$ & $\\mathrm{-}$\\\\\n\t\t\t$156$ & $65.01$ & $13.43$ & $4.84$ & $61.59$ & $10.59$ & $5.82$ & $126.33$ & $16.40$ & $7.70$\\\\\n\t\t\t$276$ & $22.17$ & $9.54$ & $2.32$ & $29.55$ & $7.95$ & $3.72$ & $51.61$ & $11.70$ & $4.41$\\\\\n\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{tab:mysobscounts}Hard, soft and full band counts for sources that can not be classified according to their dropout band. We refer to the top 5 sources as \\textit{'low significance objects'}. For objects that are not detected we give an upper limit on the counts. We set $\\sigma$ to -1.00 and mark the signal-to-noise ratio with a dash. All values were directly extracted from the \\textit{Chandra}\\ 4-Ms catalog \\protect\\citep{Xue:2011aa}.}\n\\end{table*}\n\nFive (190, 280, 333, 384, 643) of these eight objects are especially interesting since they do not have a counterpart in the optical or the infrared. We refer to these sources as \\textit{'low significance objects'}. We show these five objects in the upper panel of Figure \\ref{fig:mysteryobjects}. For three (104, 156, 276) of these eight objects it is unclear which source represents the counterpart in the optical and infrared since multiple objects are visible in the GOODS\/ACS and CANDELS bands. These objects are shown in the lower panel of Figure \\ref{fig:mysteryobjects}. \n\nOnly three of the eight objects (333, 156, 276) are detected in the full, the hard and the soft band. 104 and 280 are detected in the soft band only. 384 and 643 are found in the soft and in the full band. 190 is detected in the full band only. Only one of the low significance objects (333) and one of the three sources with multiple counterparts in the optical and infrared (156) show a signal-to-noise ratio $\\geq5$ in at least one of the bands. Out of the 371 objects with enough filter coverage and intact images 140 (37.7\\%) have a signal-to-noise ratio $\\geq5$ in the soft, hard or full band. See Table \\ref{tab:SN} for the number of sources with signal-to-noise ratios $\\geq1$ and $\\geq5$ in the individual bands.\n\nThe objects for which we do not detect a counterpart in the optical and the infrared could be spurious detections. On the one hand, \\cite{Xue:2011aa} report that, for the entire catalog, the probability of a source not being real is < 0.004. The entire catalog does therefore contain up to three spurious sources per band. We do however only consider sources that are also covered by GOODS and CANDELS. The CANDELS wide and deep survey fields are $~0.03$ $\\mathrm{deg}^2$ in size and hence only make up $\\sim27\\%$ of the CDF-S ($0.11$ $\\mathrm{deg}^2$) area. According to this, we would expect to find $\\sim0.8$ spurious detections per band. So, we can not rule out the fact that we may have found one or more false detections. On the other hand, we find many sources of comparable X-ray brightness that do possess an optical and\/or infrared counterpart and that are detected in both bands (Figure \\ref{fig:counts}). 333, 384 and 643 are also close to a bright galaxy which might be why the counterpart remains undetected. This could indicate that at least some of the sources are real.  \n\nThe Lyman Break Technique is not applicable to the low significance objects and we are unable to measure a photometric redshift without a detection in the optical and the infrared. Since 333 is detected in the hard and the soft band, it is the only object for which we can apply our Hardness Ratio test (see Section \\ref{sec:HR}). With $HR=-0.01$ 333 could be a potential high-redshift AGN candidate. A negative Hardness Ratio alone does however not convince us of 333 indeed being at high-redshift. At this point we can thus not determine if our sources are real high-redshift AGN candidates, false detections or low-redshift objects that are optically faint. Hopefully, the forthcoming 7-Ms observations of the CDF-S (PI: William Brandt, Proposal ID: 15900132) will shed more light on our five low significance objects. We eliminate all eight sources from our sample. We stress that these targets could still be high-redshift AGN.   \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=\\columnwidth]{hardsoftcounts}\n\\caption{\\label{fig:counts} Counts in the hard and in the soft band for the \\textit{Chandra}\\ 4-Ms sources. We highlight the eight objects that could not be classified according to the Lyman Break Technique. These eight sources might be real sources or spurious detections. The black points illustrate the 58 objects that are left in our sample. The grey points show the positions of the additional 502 \\textit{Chandra}\\ 4-Ms objects that are also classified as AGN. They were excluded because of insufficient filter coverage or quality or because they are clear low-redshift dropouts. \\protect\\cite{Xue:2011aa} categorize all of these eight objects and all 58 sample sources as AGN.}\n\\end{centering}\n\\end{figure}\n\nAfter discarding targets that are clearly visible in all bands and eliminating the eight objects that could not be classified according to the Lyman Break Technique, 58 $B$, $V$, $i$, $z$ and $Y$-dropouts remain in our main sample. For sources that we visually classify as $B$-dropouts the signal-to-noise in the $V$-band might be too low for a detection by SExtractor. By eliminating visually classified $B$-dropouts we could hence be missing objects that might be classified as $z\\sim5$ sources by other redshift tests. We therefore keep $B$-dropouts in our sample.  \n  \n\\subsection{Photometric Redshift measurements}\n\\label{sec:photoredcode}\nEven though the Lyman Break Technique provides a fast and easy way of identifying possible candidates, it is not without caveats. Dust in red, low-redshift galaxies can produce a sharp break in the SED that might be mistaken for the Lyman Break \\citep{Dunlop:2007aa, McLure:2010aa, Finkelstein:2012aa}. \nApplying the Lyman Break Technique therefore only produces a sample that may still contain low-redshift interlopers. \nYet, even the results of a photometric redshift code have to be treated carefully. While sources with a low photometric redshift are most likely indeed nearby objects, high photometric redshift results are less reliable. This is mainly due to template incompleteness and large photometric errors for faint high-redshift sources.   \n\nThe photometric redshifts we determine for our objects prove to be highly dependent on the filters used as input. We use the photometric redshift code EAZY \\citep{Brammer:2008aa}. We apply the default template set by \\cite{Brammer:2008aa} which is part of the EAZY distribution. Five of these six EAZY templates were created by using the \\cite{Blanton:2007aa} algorithm to reduce the template set by \\cite{Grazian:2006aa}. The sixth template describes a young starburst with a dust screen following the Calzetti law with $A_\\mathrm{v}$ = 2.75 \\citep{Calzetti:2000aa}.\nDue to large uncertainties in the luminosity function of high-redshift AGN, we do not include a luminosity prior when running EAZY.\nTo gain reliable redshift values we take the \\textit{Spitzer}\/IRAC images into account. We carefully compare the $H$ band CANDELS images to the \\textit{Spitzer}\\ 3.6 micron images to determine which of our candidates possesses a \\textit{Spitzer}\\ counterpart and for which sources the \\textit{Spitzer}\\ flux values can not be used due to source confusion. We include the \\textit{Spitzer}\\ images for 30 of our 58 objects. We show the $H$ band and \\textit{Spitzer}\\ stamps for these sources in the appendix.  We do not perform aperture photometry on the \\textit{Spitzer}\\ images, but simply extract the $1.5^{\\prime\\prime}$ aperture radius flux values from the \\textit{Spitzer}\\ catalog by \\cite{Damen:2011aa}.\n\nIf a source is not detected in a passband, we use the $1\\sigma$ sensitivity limit of this filter as an upper limit in the EAZY run. For the GOODS\/ACS bands $B$, $V$, $i$ and $z$ we determine the sensitivity limits by measuring the mean background flux. We measure the number of background counts for six of our objects. For each source, we determine the flux within apertures of varying size at five different positions. For the CANDELS bands $Y$, $J$ and $H$ we rely on the sensitivity limits reported in \\cite{Grogin:2011aa}. Table \\ref{tab:fluxlimits} shows the flux limits that we used in our analysis.\n\n\\begin{table}\n\t\\begin{center}\n\t\t\\begin{tabular}{lc}\n\t\t\t\\toprule\n\t\t\t{} & {$1\\sigma$ flux limit } \\\\\n\t\t\t{} & {in $\\mu$Jy \/ $\\mathrm{arcsec}^2$} \\\\ \n\t\t\t\\midrule\n\t\t\t$B$ & $4.636\\e{-2}$\\\\\n\t\t\t$V$ & $4.160\\e{-2}$\\\\\n\t\t\t$i$ & $8.255\\e{-2}$\\\\\n\t\t\t$z$ & $1.487\\e{-2}$\\\\ \n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{tab:fluxlimits}GOODS\/ACS flux limits. We determine these mean sensitivity limits by measuring the mean background flux for six different objects. For each object we determine the background counts within five apertures of varying size.}\n\\end{table}  \n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{paper_SED_wS_nS.pdf}\n\\caption{\\label{fig:244} Best fit SED for 244 determined by running EAZY without the \\textit{Spitzer}\\ data ($top$) and with the \\textit{Spitzer}\\ data ($bottom$). If we only use the GOODS\/ACS and CANDELS filters as input for EAZY, the photometric redshift code will classify 402 as a $z\\sim6.54$ source. Even though the flux values are fit well by this SED, the shape of the SED seems unphysical for a $z\\sim6.54$ source. We would expect the continuum flux to be lower and bluer. A comparison to the $z\\sim7$ UV luminosity function by \\protect\\cite{Bouwens:2014aa} shows that 244 would indeed be bright if it was at  $z\\sim6.54$ ($M_{H}=-24.6$). The $\\chi^2$ distribution on the right shows multiple secondary low redshift solutions. SED shape and $\\chi^2$ distribution thus suggest that 244 might be a low-redshift object. In the bottom panel we illustrate the best fit SED determined by including the \\textit{Spitzer}\\ 3.6 and 4.5 micron IRAC channels. 244 is now exposed as a low-redshift source. Based upon $z\\sim1.9$ we reject 244 as a possible high-redshift candidate. The shown limits correspond to the 1$\\sigma$ sensitivity limits. }\n\\end{figure*}\n\nIncluding the \\textit{Spitzer}\\ data turns out to be crucial for our purposes. As an example we show the photometric redshift code results we get for source 244 in Figure \\ref{fig:244}. By running EAZY without the \\textit{Spitzer}\\ flux values we gain $z_\\mathrm{phot}$ = 6.54. Assuming that this photometric redshift value is correct, we determine an absolute magnitude of $M_{H}=-24.63$ for the $H$-band. \\cite{Bouwens:2014aa} however give $M_\\mathrm{UV}^{*}=-21.2$ for $z\\sim6$ Lyman Break Galaxies. 244 would therefore be very bright if it indeed was at $z\\sim6$. Including the 3.6 and 4.5 micron \\textit{Spitzer}\\ flux values in the photometric redshift code analysis results in  $z_\\mathrm{phot}$ = 1.9. Including the \\textit{Spitzer}\\ infrared data hence proves to be crucial in revealing low-redshift interlopers.  \n\n\\subsection{Stacking GOODS\/ACS data }\nWe combine the GOODS\/ACS images of each object into stacks. We generated three stacks per source: (1) combines $B$ and $V$, (2) $B$, $V$ and $i$ and (3) $B$, $V$, $i$, $z$. We examine these deeper stacks for detections by running SExtractor on them. We use a detection threshold of 1.5 $\\sigma$ (DETECT$\\_$TRESH = 1.5) and a minimum detection area of 15 pixels above the threshold (DETECT$\\_$MINAREA = 15). The remaining SExtractor parameter values are left at their default values. A detection in the first stack indicates a source that drops out before or in the $B$-band and therefore implies $z\\lesssim4$. Sources that are detected in (1), (2) and (3) are hence of no interest to us. We assume $z\\sim5$ if an object is detected in (2) and (3), but not in (1). Only being detected in (3) indicates $z\\sim6$. Finally, no detection in any stack signals $z\\gtrsim7$. Using this approach we find 35\\ $z\\lesssim4$, 5\\ $z\\sim5$, 3\\ $z\\sim6$ and 13\\ $z\\gtrsim7$ sources. The stacking analysis proves to be inconclusive for two objects (303, 651). Source 303 is detected in (1) and (3), but not in (2). 651 is only found in stack (2).  \n\n\n\\subsection{Colour criteria}\n\\label{sec:cc}\nWe obtain an additional redshift indication by applying colour criteria based upon population synthesis models \\citep{Guhathakurta:1990aa,Steidel:1992gf}. These colour criteria do not only depend on the position of the Lyman Break, but also use the overall SED shape. The criteria we apply here are based on \\cite{Vanzella:2009aa} who used 114 Lyman Break Galaxies from the GOODS field. The sample consisted of 51 $z\\sim4$, 31 $z\\sim5$ and 32 $z\\sim6$ Lyman Break Galaxies. These objects were first chosen by applying the Lyman Break Technique and then followed up spectroscopically.  \n\nObjects that fulfill the following condition are classified as $z\\sim4$ sources:\\\\\n\\begin{equation}\n(B-V)\\geq(V-z)\\,\\wedge\\,(B-V)\\geq1.1\\,\\wedge\\,(V-z)\\leq1.6\n\\end{equation}\n\n\\noindent Here $\\wedge$ and $\\vee$ represent the logical 'and' and 'or' operators respectively. $z\\sim5$ objects need to satisfy the constraints given here:\n\n\\begin{align}\n\\begin{split}\n[(V-i) >1.5+0.9\\times(i-z)]\\vee[(V-i)>2.0]\\wedge\\\\\n(V-i)\\geq1.2\\wedge (i-z)\\leq1.3\\wedge\\mathrm{(S\/N)}_{B}<2\n\\end{split}\n\\end{align}\n\n\\noindent For $z\\sim6$ galaxies these conditions apply:\n\n\\begin{equation}\n(i-z)>1.3\\wedge[(\\mathrm{S\/N)_{B}<2\\vee(\\mathrm{S\/N)}_{V}<2}]\n\\end{equation}\nTo classify the 58 possible candidates according to colour criteria, we use the magnitude values from our aperture photometry (see \\ref{sec:aperturephot}). If a source is not detected, we use the 1$\\sigma$ sensitivity limit as an upper limit. 49\\ of the 58 sources do not fulfill any colour criteria. Note that a source that simultaneously possesses upper limits in the $B$ and $V$ or $V$ and $z$ band is not included in the $z\\sim4$ diagram since its position cannot be determined. The same applies for the $z\\sim5$ and $z\\sim6$ diagrams. Furthermore, we only use colour criteria to determine $z\\sim4$, $z\\sim5$ and $z\\sim6$ dropouts. The 49\\ objects that can not be classified are hence not all necessarily at $z < 4$. Figure \\ref{fig:cc} shows the colour-colour diagrams that illustrate these criteria. Based on colour criteria we find 2\\ $z\\sim4$ (373, 444), 2\\ $z\\sim5$ (303, 321) and 5\\ $z\\sim6$ (226, 244, 296, 522, 589) objects.\\\\\n\n\\subsection{X-rays as a photometric redshift indicator}\n\\label{sec:HR}\n\n\\begin{figure*}\n\\begin{centering}\n\\begin{minipage}[t]{0.49\\textwidth}\n\\begin{centering}\n\\includegraphics[scale=0.31]{HR_model_z.pdf}\n\\end{centering}\n\\end{minipage}\n\\begin{minipage}[t]{0.49\\textwidth}\n\\begin{centering}\n\\includegraphics[scale=0.31]{HR_z_10.pdf}\n\\end{centering}\n\\end{minipage}\n\\caption{\\label{fig:HR}AGN X-ray spectrum simulation. Shown on the $left$ is our model that we generated by running Xspec \\citep{Arnaud:1996aa}. We assume a power law with a slope of 1.8 and a turnover due to photoelectric absorption (Xspec \\texttt{zphabs*zpow} model). This left plot illustrates the model SED for moderately high column densities (line colours) at several redshifts (line types). With increasing redshift the spectrum gets shifted to lower energies and the number of counts in the hard and in the soft band changes. $Right$: measured Hardness Ratio HR as a function of redshift ($HR=\\frac{H-S}{H+S}$, $H$ and $S$ representing the counts in the hard and in the soft band respectively). Based on our model, a HR > 0 indicates $z < 4.34$ for column densities up to $N_\\mathrm{H}$\\ = $10^{23.5} \\mathrm{cm}^{-2}$. In our analysis we thus dismiss objects with an HR > 0 as obscured AGN at low redshift.}\n\\end{centering}\n\\end{figure*}\n\n\nThe Hardness Ratio (HR), sometimes denoted as X-ray colour, represents an additional indicator for high-redshift AGN. The Hardness Ratio is defined as: \n\\begin{equation}\\label{eq:HR}\nHR=\\frac{H-S}{H+S}\n\\end{equation} \nHere H and S represent the observed frame hard (2 - 10 keV) and soft (0.5 - 2 keV) band counts respectively.\n\nTo zeroth-order, an AGN X-ray spectrum follows a powerlaw with an obscuration dependent turnover at lower energies. In the Compton-thin regime a higher column density $N_\\mathrm{H}$\\  means that, relative to the hard band, we will detect fewer counts in the soft band. In a galaxy's restframe the X-ray spectrum hence appears harder for higher obscuration. With increasing redshift the spectrum gets shifted to lower energies and the number of counts that we observe in the hard and in the soft band changes accordingly. We can therefore use the Hardness Ratio as an additional redshift indicator.\n\nA soft X-ray spectrum is then expected for Compton-thin ($N_\\mathrm{H}$\\ $< 10^{24}\\mathrm{cm}^{-2}$) objects. For sources in which $10^{24}\\lesssim$ $N_\\mathrm{H}$\\ $\\mathrm{H}\\lesssim 10^{25}\\mathrm{cm}^{-2}$, i.e. transmission-dominated Compton-thick AGN, the direct emission is still visible, although the E $<$ 10~keV radiation is completely obscured by photoelectric absorption, while the detected emission at higher energies is reduced by Compton scattering \\citep{Comastri:2010aa, Murphy:2009aa}. Hence, in these cases we expect to observe a hard X-ray spectrum even for sources at $z > 5$. When $N_\\mathrm{H}$\\ $ > 10^{25}\\mathrm{cm}^{-2}$, i.e. for reflection-dominated Compton-thick AGN, we only observe the small fraction of the initial emission which is reflected off the accretion disk or the obscuring material \\citep{Ajello:2009aa,2012MNRAS.422.1166B}. We can hence not make the same assumptions as for Compton-thin sources. Nevertheless, it is safe to assume that all $z\\gtrsim5$ sources that can be detected individually in the X-rays are Compton-thin objects. Compton-thick $z\\gtrsim5$ AGN would simply be too faint to be detected individually and are therefore not part of our sample.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.49\\textwidth]{HR_hist}\n\\end{centering}\n\\caption{\\label{fig:HRhist}Number of objects per HR bin. This figure illustrates the distribution of our 54 objects in terms of HR. HR < 0  indicates that this source might be at $z > 4.34$. A positive HR suggests that this source might be at $z\\leq$ 4.34. Thus ten of our 54 objects might be at $z > 4.34$. Not shown are objects 496 and 578. These two sources are only detected in the full band and can therefore not be constrained by the Hardness Ratio. The numbers in each bar coincide with the object IDs in the corresponding HR bin.}\n\\end{figure}\n\nTo quantify the HR($z$) relation we use the X-ray spectral fitting tool Xspec to simulate X-ray spectra at different redshifts \\citep{Arnaud:1996aa}. We use a \\texttt{zphabs*zpow} model and assume a power law slope of 1.8 \\citep{Turner:1997aa, Tozzi:2006aa}. Figure \\ref{fig:HR} summarizes our results. The left panel illustrates our model for $N_\\mathrm{H}$\\ $ = 10^{22} \\mathrm{cm}^{-2}$, $10^{22.5} \\mathrm{cm}^{-2}$ and $10^{23.5} \\mathrm{cm}^{-2}$  (blue, yellow and green, respectively) at $z$ = 0.1, 3 and 6 (dot dashed, solid and dashed line, respectively). It is evident that the number of counts in the soft and hard band changes with redshift. After including the \\textit{Chandra}\\ Redistribution Matrix (RMF) and Auxiliary Response Files (ARF) for on-axis sources, we measure the spectral counts in the hard and in the soft band and determine the HR. The right panel of Figure \\ref{fig:HR} shows our results. According to our simulations HR > 0 signifies $z < 4.34$ for sources with $N_\\mathrm{H}$\\ up to $10^{23.5} \\mathrm{cm}^{-2}$. Allowing for a small amount of transmission, e.g. 1$\\%$ by using a \\texttt{zpcfabs*zpow} model does not change our results significantly (HR = 0 for $z$ = 4.3). In our analysis we hence discard objects with HR > 0. Figure \\ref{fig:HRhist}, which illustrates our results, shows that based solely on the HR, ten of our 58 candidates might be at $z > 4.34$. For 496 and 583 the HR can not be used to constrain $z$ since they are only detected in the full band. Table \\ref{tab:counts} summarizes the X-ray counts, signal-to-noise ratios and Hardness Ratios for each of our main sample sources.\n\nOur results are in good agreement with a similar analysis by \\cite{Wang:2004aa}. They also used Xspec \\citep{Arnaud:1996aa} to simulate the HR at different redshifts. Yet, \\cite{Wang:2004aa} chose a power law slope of 1.9 and used a \\texttt{wabs} model. Our analysis shows that $z > 5$ sources with $N_\\mathrm{H}$\\ up to $10^{23} \\mathrm{cm}^{-2}$ should have a HR of $\\sim-0.3$, \\cite{Wang:2004aa} find HR$\\sim-0.5$. For $N_\\mathrm{H}$\\ = $10^{23} \\mathrm{cm}^{-2}$ we determine HR = 0 at $z\\sim2$, \\cite{Wang:2004aa} find HR = 0 at $z\\sim1.5$.  \n\n\\newcommand\\low{\\textcolor{red}{$\\times$}}\n\\newcommand\\high{\\textcolor{green}{\\checkmark}}\n\\definecolor{green}{HTML}{3D9D93}\n\\definecolor{red}{HTML}{A75379}\n\n\\begin{table*}\n\t\\begin{center}\n\t\t\\begin{tabular}{llccccc}\n\t\t\\toprule\n\t\t\t{} & {ID} & {visual classification} & {colour criteria} & {Stacking} & {Hardness Ratio} & {photo-z}\\\\\n\t\t\t\\midrule\n\t\t\t{} & $121$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $150$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $173$ & \\low & 0 & \\low & \\high & \\low \\\\\n\t\t\t{} & $184$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $189$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $199$ & \\high & 0 & \\high & \\high & \\low \\\\\n\t\t\t{} & $211$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $217$ & \\high & 0 & \\high & \\high & \\low \\\\\n\t\t\t{} & $221$ & \\low & 0 & \\low & \\low & \\low\\\\\n\t\t\t{} & $226$ & \\low & \\high & \\low & \\low & \\high \\\\\n\t\t\t{} & $242$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $244$ & \\high & \\high & \\high & \\low & \\low \\\\\n\t\t\t{} & $258$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $273$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $296$ & \\high & \\high & \\high & \\low & \\low \\\\\n\t\t\t{} & $301$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $302$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $303$ & \\high & \\high & 0 & \\low & \\low \\\\\n\t\t\t{} & $306$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $318$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $321$ & \\low & \\high & \\low & \\low & \\low \\\\\n\t\t\t{} & $325$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $328$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $331$ & \\low & 0 & \\low & \\low & \\low\\\\\n\t\t\t{} & $348$ & \\high & 0 & \\high & \\low & \\low \\\\\n\t\t\t{} & $354$ & \\high & 0  & \\high & \\high & \\low \\\\\n\t\t\t{} & $371$ & \\high & 0  & \\high & \\low & \\high \\\\\n\t\t\t{} & $373$ & \\low & \\low  & \\low & \\low & \\low \\\\\n\t\t\t{} & $389$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $392$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $402$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $403$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $410$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $428$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $430$ & \\high & 0 & \\high & \\low & \\high \\\\\n \t\t\t{} & $444$ & \\low & \\low  & \\low & \\low & \\low \\\\\n\t\t\t{} & $455$ & \\low & 0  & \\low & \\low & \\low\\\\\n\t\t\t$\\rightarrow$ & $456$ & \\high & 0  & \\high & \\high & \\high \\\\\n\t\t\t{} & $460$ & \\high & 0  & \\high & \\high & \\low \\\\\n\t\t\t{} & $462$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $466$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $485$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $496$ & \\low & 0  & \\low & 0 & \\low \\\\\n\t\t\t{} & $522$ & \\low & \\high  & \\high & \\high& \\low \\\\\n\t\t\t{} & $535$ & \\low & 0  & \\low & \\high & \\low \\\\\n\t\t\t{} & $539$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $546$ & \\low & 0  & \\low & \\low & \\low\\\\\n\t\t\t{} & $556$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $574$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t$\\rightarrow$ & $578$ & \\high & 0  & \\high & 0 & \\high \\\\\n\t\t\t$\\rightarrow$ & $583$ & \\high & 0  & \\high & \\high & \\high \\\\\n\t\t\t{} & $589$ & \\low & \\high & \\low & \\low & \\low \\\\\n\t\t\t{} & $591$ & \\low & 0 & \\low & \\low & \\low \\\\\n\t\t\t{} & $620$ & \\low & 0  & \\low & \\high & \\low \\\\\n\t\t\t{} & $624$ & \\high & 0  & \\high & \\low & \\low \\\\\n\t\t\t{} & $625$ & \\low & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $630$ &  \\high & 0  & \\low & \\low & \\low \\\\\n\t\t\t{} & $651$ & \\high & 0 & 0 & \\low & \\low \\\\\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{center}\n\\caption{\\label{tab:allresults} Overview showing all five redshift test results. Sources for which the test indicates $z < 5$ are marked with \\low. If the redshift test results in $z\\geq$ 5, we show a \\high. Objects that could not be classified via colour criteria, stacking or the Hardness Ratio are marked with 0. For the Hardness Ratio we can only distinguish between $z < 4.3$ (HR > 0) and $z\\geq$ 4.3 (HR $\\leq$ 0). Hence, for the Hardness Ratio \\high means $z\\geq$ 4.3. After combining all redshift tests we are left with three possible candidates (arrows). See Table \\ref{tab:overview} and Table \\ref{tab:counts} for a detailed overview of the results and the X-ray counts of each individual source.}\n\\end{table*}\n\\newpage\n\n\\section{Combining all redshift tests}\n\\label{sec:combination}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{paper_HR_photz_chi2}\n\\caption{\\label{fig:HR_photz_chi2}After we combine our visual classifications with our $z_\\mathrm{phot}$, stacking and colour criteria results, we are left with seven possible candidates. We show their Hardness Ratio and photometric redshift values in the $left$ panel. The $right$ figure illustrates their $z_\\mathrm{phot}$ $\\chi^2$ distributions. By also eliminating objects with HR > 0, we are left with three final candidates (456, 578, 583). The left panel does not show sources 496 and 578. These two objects are only detected in the full band and can therefore not be constrained by the Hardness Ratio. While we find a low $z_\\mathrm{phot}$ for 496, 578 has $z_\\mathrm{phot}=6.77$ and is therefore one of our final high $z$ candidates. We show $1\\sigma$ error bars. For the X-ray counts and signal-to-noise ratios of each individual source see Table \\ref{tab:counts}.}\n\\end{figure*}\n\nWe now combine our stacking, colour criteria and photometric redshift code results. We exclude objects with  $z_\\mathrm{phot}<5$, $z_{\\mathrm{stacking}}<5$ and $z_{\\mathrm{colour}}<5$. Without the Hardness Ratio constraint six $z>5$ objects remain (Figure \\ref{fig:HR_photz_chi2}). Neither stacking, colour criteria nor our visual classification contradict $z>5$ for these six sources (226, 371, 456, 578, 583, 430). For 371, 456, 578, 583 and 430 the $1\\sigma$ error bars on $z_\\mathrm{phot}$ are asymmetric and reach below $z\\sim5$. The $\\chi^{2}$ distributions (Figure \\ref{fig:HR_photz_chi2}) also do not show clear global minima. For 430 and 578 the $\\chi^2$ distributions are flat and allow for a wide range of lower redshift solutions. The $z_\\mathrm{phot}$ solution for 456 is at the upper end of our allowed $z_\\mathrm{phot}$ range ($z_\\mathrm{phot}=$ 0 to 11). 371 shows additional minima at $z_\\mathrm{phot}\\sim1$. For 226 we determine an absolute magnitude of $M_H = -25.6$ assuming a photometric redshift of 5.43. 226 would therefore be extremely bright if it indeed was at $z \\sim 5$ (see section \\ref{sec:photoredcode}). There is thus only little evidence supporting the fact that these objects might be at high redshift. Only 226, 456 and 583 show global minima and are thus our most promising candidates.\n\n\\subsection{Our three final candidates}\n\\label{sec:final_three}\n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{456_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{578_series_stamps}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width=\\textwidth]{583_series_stamps}\n\\end{minipage}\n\\caption{\\label{fig:candidates}Candidate $z > 5$ AGN that remain after combining all redshift tests. 456, 578 and 583 are the only sources that remain after we combine our stacking ($z_{\\mathrm{stacking}}\\sim7$ for all), colour criteria (all not classified), Hardness Ratio (456: $HR=-0.289$, 578: not classified, 583: $HR=-0.511$ ) and photometric redshift code results (456: $z_{\\mathrm{phot}}=10.913$, 578: $z_{\\mathrm{phot}}=6.766$, 583: $z_{\\mathrm{phot}}=9.364$). Visually we classify 456, 578 and 583 as $z$ dropouts. Due to source confusion, we do not use the \\textit{Spitzer}\\ 3.6 and 4.5 micron images when running the photometric redshift code for these objects. All images are colour inverted and are $10^{\\prime\\prime}$ x $10^{\\prime\\prime}$ in size.}\n\\end{figure*}\n\nWe now also take the Hardness Ratio information into account and exclude three of the six remaining sources based on a positive HR value. After combining all of our redshift tests we are hence left with three final high-redshift candidates (456, 578, 583, Figure \\ref{fig:candidates}). 456 and 583 have a negative Hardness Ratio whereas 578 cannot be constrained by HR since it is only detected in the full band.  \n\nFor 456 we find $z_\\mathrm{phot} = 10.91^{-0.92}_{-8.39}$ ($\\chi^2 \\sim 0$) and HR = -0.29. For 578 we determine $z_\\mathrm{phot} = 6.77^{+0.97}_{-5.10}$ ($\\chi^2 \\sim 0$). 583 has HR = -0.51 and $z_\\mathrm{phot}= 9.36^{+0.63}_{-4.82}$ ($\\chi^2 \\sim 0$). 456, 578 and 583 could not be classified according to our colour criteria. Nonetheless, our visual classification ($z$ dropouts) and our stacking analysis ($z_{\\mathrm{stacking}}\\sim7$ for all) indicate that these sources might be $z > 5$ AGN.   \n\n\\subsection{Expected number densities}\n\nWe note that 456, 578 and 583 seem bright for $z\\sim7$ and $z\\sim10$ sources. We use UV rest-frame galaxy luminosity functions by \\cite{Bouwens:2014aa} to quantify this statement. To justify our comparison to the galaxy, and not quasar luminosity function, we calculate $\\alpha_\\mathrm{ox}$ for our candidates. $\\alpha_\\mathrm{ox}$, the X-ray to optical-UV ratio, is defined as $\\alpha_\\mathrm{ox}=\\log\\big[{L_{2500\\AA}\/L_{2keV}}\\big]\/2.605$ (e.g. \\citealt{Tananbaum:1979aa, Wilkes:1994aa, Vignali:2003aa, Steffen:2006aa}). \\cite{Lusso:2010aa} analyse a sample of 545 X-ray selected Type 1 AGN from the XMM-COSMOS survey and find a mean $\\alpha_\\mathrm{ox}$ value of $\\sim1.37$ and a weak redshift dependence out to $z\\sim4$. We use the $H$-band and soft band flux values to calculate $\\alpha_\\mathrm{ox}$ for 456, 578 and 583. We find $\\alpha_{\\mathrm{ox}}=0.11$, $0.35$ and $-0.02$ for 456, 578 and 583, respectively. Assuming that $\\alpha_\\mathrm{ox}$ has no strong redshift dependence for $z>4$, this indicates that 456, 578 and 583 are not high-redshift quasars and justifies our comparison to the results by \\cite{Bouwens:2014aa}.     \n\nWe estimate the number of sources similar to 456, 578 and 583 that we expect to find in our field. Since we only analyse objects for which GOODS\/ACS and CANDELS data is available, we are only considering the area of the CANDELS deep and wide surveys (0.03 $\\mathrm{deg}^2$) and not the entire CDF-S (0.11 $\\mathrm{deg}^2$). Based on the surface density of $z\\sim7$ and $z\\sim10$ galaxies by \\cite{Bouwens:2014aa}, we expect to find $0.14\\pm0.14$ sources as bright as 456 at $z\\sim10$, $3.72\\pm0.80$ objects similar to 578 and less than $0.15$ $z\\sim10$ objects as bright as 583. We hence expect high-redshift sources as bright as 456, 578 and 583 to be rare. \n\n\\subsection{Deeper Y-band imaging for 583}\n\\label{sec:HUDF}\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width = \\textwidth]{paper_456}\n\\end{minipage}\n\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width = \\textwidth]{paper_578}\n\\end{minipage}\n\n\\caption{\\label{fig:456_578}Photometric redshift code results for objects 456 and 578 before ($top\\ panels$) and after including the HUGS $K_S$-band ($bottom\\ panels $). After combining all of our redshift tests, we are left with three final candidates (for 583 see Figure \\ref{fig:583}). For these we include the Hawk-I UDS and GOODS Survey (HUGS) \\protect\\citep{Fontana:2014aa} $K_S$-band values when determining the photometric redshift. We show the best fit template SEDs ($left$), the $\\chi^2$ distributions ($middle$) and the corresponding images ($right$), before and after including the $K_S$-band. For 456 we redetermine the $J$-band upper limit. Our result is close to the value reported by \\protect\\cite{Grogin:2011aa} and does not change the $z_\\mathrm{phot}$ result significantly. For both, 456 and 578, a low redshift solution is allowed within the $2\\sigma$ error. These two final candidates are hence likely to be at low redshift. We use the $\\Delta\\chi^2$ method under the assumption of one free parameter ($z_\\mathrm{phot}$) to compute the $1, 2$ and $3\\sigma$ error bars that are shown in the middle panels. All images are colour inverted and are $5^{\\prime\\prime} \\times 5^{\\prime\\prime}$ in size.} \n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\includegraphics[width = \\textwidth]{paper_583}\n\\end{minipage}\n\\caption{\\label{fig:583} Photometric redshift code results for object 583 before ($top\\ panels$) and after including the HUDF $Y$-band ($middle\\ panels$) and the HUGS $K_S$-band ($bottom\\ panels$), analogue to Figure \\ref{fig:456_578}. After combining all of our redshift tests, we are left with three final candidates (for 456 and 578 see Figure \\ref{fig:456_578}). Fortunately, 583 is covered by the HUDF. We thus perform aperture photometry on the deep HST\/WFC3 HUDF $Y$-band and replace the CANDELS $Y$-band value with this flux ($middle\\ panels$). We also include the Hawk-I UDS and GOODS Survey (HUGS) \\protect\\citep{Fontana:2014aa} $K_S$-band value when determining the photometric redshift ($bottom\\ row$). With the HUDF $Y$-band and the HUGS $K_S$-band 583's photometric redshift value drops to $2.68$. It is hence most likely a low-redshift source. We show the best fit template SEDs ($left$), the $\\chi^2$ distributions ($middle$) and the corresponding images ($right$). We use the $\\Delta\\chi^2$ method under the assumption of one free parameter ($z_\\mathrm{phot}$) to compute the shown $1, 2$ and $3\\sigma$ error bars. All images are colour inverted and are $5^{\\prime\\prime} \\times 5^{\\prime\\prime}$ in size.}\n\\end{figure*}\n  \n583, one of our three final candidates, has a high photometric redshift ($z_\\mathrm{phot} = 9.36$) and shows a clear global minimum in the $\\chi^2$ distribution. In terms of the $\\chi^2$ distribution it is thus our most promising candidate. Fortunately, 583 is part of the $Hubble$ Ultra Deep Field (HUDF) \\citep{Beckwith:2006aa}. We are especially interested in deeper $Y$-band imaging since 583's high photometric redshift hinges on the upper limit in this band. We show our results in Figure \\ref{fig:583}. The top panels illustrate, that the upper limit in the $Y$-band, for which we used the official sensitivity limit given by \\cite{Grogin:2011aa}, is very low and thus forces a strong break in 583's SED. We hence combine all available HST\/WFC3 HUDF $Y$-band images (\\citealt{Koekemoer:2013aa}, HST Program ID 12498, PI: R. Ellis; \\citealt{Illingworth:2013aa}, HST Program ID 11563, PI: G. Illingworth, HST Program ID 12099, PI: A. Riess) using the HST DrizzlePac package. We measure the flux of 583 using simple aperture photometry and a $0.3^{\\prime\\prime}$ radius aperture (middle panels) and rerun EAZY. We use the $\\Delta\\chi^2$ method under the assumption of one free parameter ($z_\\mathrm{phot}$) to compute the $1, 2$ and $3\\sigma$ error bars. We still determine a high photometric redshift of $z_\\mathrm{phot}=7.41$, a low redshift solution at $z_\\mathrm{phot}\\sim2$ is however allowed within $1\\sigma$. 583 is hence unlikely to be at high redshift.  \n\n\\subsection{HUGS $K_S$-band data}\n\\label{sec:HUGS}\n\nTo gain more reliable photometric redshift values for all three of our final candidates, we now also take VLT\/Hawk-I $K_S$-band data into account. We take the $K_S$-band flux values for 456, 578 and 583 from the Hawk-I UDS and GOODS Survey (HUGS) catalog (v1.1) \\citep{Fontana:2014aa} and rerun EAZY. Our results are illustrated in Figures \\ref{fig:456_578} and \\ref{fig:583}. The photometric redshift value for 456 remains at $10.91^{-0.92}_{+8.39}$ ($\\chi^2 \\sim 0$). However, the $z_\\mathrm{phot}$ value seems highly dependent on the $J$-band upper limit. We thus redetermine the $J$-band sensitivity limit by measuring the background flux within 35 apertures which are scattered across the CANDELS field. We determine a flux limit close to the value reported by \\cite{Grogin:2011aa} (measured: $2.62 \\times 10^{-2} \\mu$Jy, reported: $1.97 \\times 10^{-2} \\mu$Jy for a $0.5^{\\prime\\prime}$ radius aperture). We rerun EAZY and find a value almost identical to the previous result ($z_\\mathrm{phot} = 10.91^{-1.14}_{-7.11}$, $\\chi^2 = 0.01$). 456 thus remains a source with a high photometric redshift. The $2\\sigma$ error bar, computed through the $\\Delta\\chi^2$ method, does however allow for a low-redshift solution at $z\\sim4$. For 578 the $K_S$-band causes the photometric redshift to increase from 6.77 to $7.49^{+0.54}_{-4.60}$ ($\\chi^2 = 0.08$). Nonetheless, similar to 456, 578's $2\\sigma$ error bar permits a $z\\sim3$ solution. For 583 we determine $z_\\mathrm{phot} = 2.68^{+5.61}_{+0.37}$ ($\\chi^2=0.21$) by including the HUDF deep $Y$-band and the HUGS $K_S$-band flux values. This $z_\\mathrm{phot}$ value matches what has previously been reported by \\cite{Szokoly:2004aa} for the galaxy next to 583 ( 53.1833, -27.7764). We thus suspect, that 583 might be part of a large clumpy galaxy at low redshift \\citep{Schawinski:2011aa}. With the X-ray emission being offset from what could be the main galaxy, this does remain a very interesting object. \n\nIn summary, the HUGS $K_S$-band and HUDF $Y$-band data causes the photometric redshifts of our three final candidates to either drop to low-redshift or to allow for a low-redshift solution within a $2\\sigma$ error bar. Considering these photometric redshift code results and how rare objects as bright as our candidates should be at high-redshift, we conclude that 456, 578 and 583 are unlikely to be at high redshift. They do however remain compelling candidates for follow-up observations. 456 remains a very interesting candidate since the $1\\sigma$ $z_\\mathrm{phot}$ error bar does not allow for lower redshift solutions. 578 could still be at high redshift and 583 is intriguing since the X-ray emission seems to be offset from the main galaxy. \n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\begin{figure*}\n\\begin{centering}\n\\includegraphics[width=\\textwidth]{four_red_all_bands}\n\\end{centering}\n\\caption{\\label{fig:BHmasses}Expected X-ray flux as a function of BH mass, Eddington ratio and redshift. We assume spherical accretion, $N_\\mathrm{H}$\\ $=10^{22}\\mathrm{cm}^{-2}$ and a bolometric correction of $k_\\mathrm{corr}=25$ to calculate the expected X-ray flux for BH masses ranging from $10^{6}\\ M_{\\odot}$ to $10^{9}\\ M_{\\odot}$, Eddington ratios between 0.01 and 1. and redshifts between $z=4$ to $z=10$. The dashed lines show the \\textit{Chandra}\\ 4-Ms flux limits in the soft, full and hard bands. The BH mass in a typical Lyman Break Galaxy at $z\\sim5$ is $\\sim10^{7}$$M_{\\odot}$. This figure illustrates that we are sensitive enough to detect such sources if we assume low obscuration and an Eddington ratio > 0.1. }\n\\end{figure*}\n\n\\subsection{Sensitivity and AGN number density}\nIn this section we show that our analysis should have been sensitive enough to detect an active BH in a typical high-redshift Lyman Break Galaxy and why we expected to find at least some high $z$ AGN. Furthermore, we determine an upper limit on the AGN number density. \n\nIn a first step we determine the BH masses and accretion rates we are sensitive to by calculating the expected X-ray flux as a function of BH mass, Eddington ratio  and redshift. We calculate the X-ray luminosity and translate it to observed flux. We use Eddington ratios between 0.01 and 1. and a redshift range from $z=4$ to $z=10$. For simplicity we assume a constant bolometric correction of $k_\\mathrm{corr}=25$ ($L_\\mathrm{bol}=k_\\mathrm{corr} L_\\mathrm{X}$, \\citealt{Vasudevan:2009aa}) and $N_\\mathrm{H}$\\ $=10^{22}\\mathrm{cm}^{-2}$. Figure \\ref{fig:BHmasses} illustrates our results. We also show the \\textit{Chandra}\\ 4-Ms flux limits which lie at $9.1\\times10^{-18}\\ \\mathrm{erg}\\ \\mathrm{s}^{-1}\\ \\mathrm{cm}^{-2}$ for the soft band and at $5.5\\times10^{-17}\\ \\mathrm{erg}\\ \\mathrm{s}^{-1}\\ \\mathrm{cm}^{-2}$ for the hard band \\citep{Xue:2011aa}. At $z\\sim5$ we are sensitive to luminosities as low as $\\sim10^{42}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ in the soft band and $\\sim10^{43}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ in the hard band.\n\nIn the GOODS field, the typical stellar mass of a Lyman Break Galaxy at $z\\sim5$ is $2.82\\times10^{10}$$M_{\\odot}$\\ \\citep{Lee:2012aa}. If we use the local BH-stellar mass relation ($M_{\\star, \\mathrm{total}}{\/}$ $M_\\mathrm{BH}$\\ $=562$, \\citealt{Jahnke:2009aa}) to determine the corresponding BH mass, we find $M_\\mathrm{BH}$\\ $=5\\times10^{7}$$M_{\\odot}$. Figure \\ref{fig:BHmasses} shows that, assuming low obscuration and Eddington ratios $>0.1$, we should have been capable of detecting such AGN at $z>5$.\n\nTo estimate the number of high $z$ AGN in the CDF-S we use the results by \\cite{Bouwens:2014aa}. They find 680 $z\\sim5$, 252 $z\\sim6$ and 113 $z\\sim7$ Lyman Break Galaxies in the CANDELS\/Deep and CANDELS\/Wide surveys for the GOODS-S field. \\cite{Nandra:2002aa} report an AGN fraction of $3\\%$ for $z\\sim3$ Lyman Break Galaxies in the Hubble Deep Field North. Assuming that this fraction does not evolve with redshift, we would expect the volume we looked at to contain $\\sim20$ AGN at $z\\sim5$, $\\sim8$ at $z\\sim6$ and $\\sim3$ at $z\\sim7$. Since we are sensitive enough to detect Compton-thin AGN in typical Lyman Break Galaxies, we would have expected to find at least some convincing high-redshift sources. Even if all of our three final candidates (456, 578, 583) would prove to be at high-redshift, the number of high-redshift AGN in our field would still be lower than expected. \n\nWe use the estimated number of high redshift AGN to determine an upper limit on the AGN number density. The combined CANDELS\/Wide and CANDELS\/Deep survey covers an area of 0.03 $\\mathrm{deg}^{2}$. At $z\\sim5$ the faint X-ray selected AGN number density is hence fewer than 655 AGN per $\\mathrm{deg}^{2}$. At $z\\sim6$ and $z\\sim7$ we would expect to find fewer than 262 and 98 AGN per $\\mathrm{deg}^{2}$, respectively. Here, we assumed the $z\\sim3$ Lyman Break Galaxy AGN fraction by \\cite{Nandra:2002aa}. This AGN fraction could however be especially low for high redshift Lyman Break Galaxies or generally evolve with redshift. We discuss this in more detail in Section \\ref{sec:explanations}.\n\n\\subsection{Comparison to existing work}\n\nWe compare our results to \\cite{Vito:2013aa}. They found three $z > 5$ AGN (139, 197, 485) in the CDF-S. Source 139 has a photometric redshift of $z_\\mathrm{phot}$ = 5.73 based on \\cite{Luo:2010aa}. Since 139 is not covered by $B$, $Y$, $J$ and $H$ it was not part of our analysis. Object 197 has $z_\\mathrm{phot}$ = 6.07 with a secondary solution at $z_\\mathrm{phot}$ = 4.39 \\citep{Luo:2010aa}. 197 is not in the GOODS field and was therefore immediately excluded by us. Source 485 has a photometric redshift of $z_\\mathrm{phot}$ = 7.62 with a secondary solution at $z_\\mathrm{phot}$ = 3.31 based on  \\cite{Luo:2010aa} and has $z_\\mathrm{phot}$ = 4.42 according to \\cite{Santini:2009aa}. Both, \\cite{Luo:2010aa} and \\cite{Santini:2009aa}, did not take CANDELS WFC3\/IR data into account when determining photometric redshifts. Instead they used the GOODS - MUSIC catalog \\citep{Grazian:2006aa} which contains the VLT\/ISAAC $J$, $H$, and $K$ bands. Taking CANDELS and \\textit{Spitzer}\\ data into account, we determined $z_\\mathrm{phot} = 2.83^{+0.54}_{-0.68}$ for object 485.\n\n\\cite{Vito:2013aa} also report spectroscopic and photometric redshifts for 31 additional AGN at lower redshifts. Seven of these 31 sources are also part of our sample. For three objects our photometric redshifts lie within $1\\sigma$ from the redshift reported by \\cite{Vito:2013aa} (331: $z_\\mathrm{phot,\\\/Vito} = 3.780$, $z_\\mathrm{phot} = 4.32^{+0.02}_{-3.63}$; 371: $z_\\mathrm{phot,\\\/Vito} = 3.10$, $z_\\mathrm{phot} = 5.56^{+0.04}_{-4.50}$; 546: $z_\\mathrm{spec,\\\/Vito} = 3.06$, $z_\\mathrm{phot} = 3.15^{+0.18}_{-0.25}$). For four sources our photometric redshifts do not lie within $1\\sigma$ (150: $z_\\mathrm{phot,\\\/Vito} = 3.34$, $z_\\mathrm{phot} = 3.63^{+0.14}_{-0.16}$; 403: $z_\\mathrm{spec,\\\/Vito} = 4.76$, $z_\\mathrm{phot} = 4.22^{+0.17}_{-0.12}$; 556: $z_\\mathrm{phot,\\\/Vito} = 3.53$, $z_\\mathrm{phot} = 2.68^{+0.17}_{-0.16}$; 651: $z_\\mathrm{phot,\\\/Vito} = 4.66$, $z_\\mathrm{phot} = 1.90^{+0.41}_{-0.34}$). The remaining 24 low-redshift AGN were not part of our sample because they were not covered by enough bands ($B$, $V$, $i$, $z$, $J$, $H$), because they were clearly visible in all bands and therefore discarded as $z<5$ sources or because their images were disturbed by artefacts.     \nExcept for source 485 and the four low-redshift AGN (150, 403, 556, 651) our findings do hence agree with the results by \\cite{Vito:2013aa}.\n\n\n\\cite{T13} searched for X-ray emission of $z = 6 - 8$ Lyman Break dropout and photometrically selected sources.  None of the $\\sim600$ $z\\sim6$, $\\sim150$ $z\\sim7$ or $\\sim80$ $z\\sim8$ sources could be detected individually in the X-rays. Stacking the X-ray data in redshift bins did not generate a significant detection either. In the stacks the $3\\sigma$ upper limits on the X-ray emission lay at $\\sim10^{41}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ (soft) and $\\sim10^{42}\\mathrm{erg}\\ \\mathrm{s}^{-1}$ (hard) for the $z\\sim6$ and $z\\sim7$ bins. Assuming a bolometric correction of $k_\\mathrm{corr}=25$ \\citep{Vasudevan:2009aa} and an Eddington ratio of 0.1, these upper limits correspond to $\\sim10^{5}$ $M_{\\odot}$\\ (soft) and $\\sim10^{6}$ $M_{\\odot}$\\ (hard) in terms of BH mass. \nIn comparison to our work, \\cite{T13} based their search on a sample of optically selected sources, whereas we selected our objects in the X-rays. Nonetheless, the results are in agreement. \n\n\\subsection{$z\\gtrsim5$ AGN in the CDF-S}\n\nIn this work we searched for possible $z > 5$ AGN candidates in the CDF-S. In contrast to \\cite{T13} we started out with a sample of confirmed X-ray sources. We used visual classification, colour criteria, stacking, a photometric redshift code and the Hardness Ratio to obtain multiple redshift indications. After dismissing sources for which our redshift tests indicated $z < 5$, three final candidates with $z_\\mathrm{phot} \\sim 7$ (578), $z_\\mathrm{phot} \\sim 9$ (583) and $z_\\mathrm{phot} \\sim 10$ (456) remained in our sample. Our comparison to the galaxy luminosity function showed that $z\\sim7$, $z\\sim9$ and $z\\sim10$ objects as bright as 578, 583 and 456 are rare. By including the Hawk-I UDS and GOODS Survey (HUGS) $K_S$-band \\citep{Fontana:2014aa} and the HUDF $Y$-band data, the photometric redshifts for our three final candidates either dropped to low-redshift (583) or allowed for a low-redshift solution within $2\\sigma$ error bars (578, 583). Our three final candidates did hence not pass this extended redshift test. We conclude that considering our photometric redshift code results and the rarity of such high-redshift objects, 456, 578 and 583 are most likely low-redshift sources. We also found five low significance objects. These sources are detected in the X-rays, but they do not seem to possess a counterpart in the optical or infrared (including the \\textit{Spitzer}\/IRAC 3.6 and 4.5 micron channels and the VLT\/Hawk-I $K_S$-band). The currently available data does not allow us to constrain their redshifts or determine if these objects are spurious detections.  \n\nBased upon currently available GOODS\/ACS, CANDELS and \\textit{Spitzer}\\ data, the analysis did therefore yield three final $z > 5$ candidates and five low significance objects for this deep, but narrow field. Including HUGS and HUDF data did however show, that 456, 578 and 583 are likely to be at low redshift. Follow-up observations are necessary to gain more reliable redshifts for our three final candidates and to constrain the nature of our five low significance objects. \n\n\\subsection{Possible explanations}\n\\label{sec:explanations}\n\nBoth, the approach by \\cite{T13} and our approach, assumed that X-ray emission is a valid tracer for BH growth. If at high redshift BHs primarily grow through BH mergers instead of accretion, electromagnetic radiation might not be emitted during the growth process. X-ray emission might hence not be a indicator for active BHs.\n\nAt high redshift the number of actively accreting BHs could also be generally low. This could be caused by a low BH occupation fraction, a low AGN fraction or BH growth through short, super-Eddington episodes. We stress the difference between the BH occupation fraction and the AGN fraction since they describe different scenarios. \n\nIf the BH occupation fraction is low only very few haloes are seeded with BHs. Our sample could therefore be too small to not only contain a BH, but to contain a BH that is also actively accreting. \n\nEven if the BH occupation fraction is high, the AGN fraction could still be low. So, even if there are plenty of BHs in our field, only few of them could be active. For instance, BHs at high redshift could only grow in optically faint galaxies. In our analysis these faint galaxies could correspond to the low significance objects that do not seem to possess an optical or infrared counterpart (190, 280, 333, 384, 643, see Section \\ref{sec:LBT}). The currently available data does not allow us to investigate these sources further. We are however hopeful that the forthcoming 7-Ms survey for the CDF-S (PI: William Brandt, Proposal ID: 15900132) will show if these objects are real.  \n\nBHs could also grow through short, super-Eddington accretion phases \\citep{Madau:2014aa, Volonteri:2014aa}. \\cite{Madau:2014aa} illustrated that a duty cycle of $20\\%$ is enough to grow a  $100$ $M_{\\odot}$\\ non-rotating seed BH into a $\\sim10^9$ $M_{\\odot}$\\ BH by $z\\sim7$. This could, for instance, be realized through five $20\\ \\mathrm{Myr}$ long growth episodes with $\\dot{m}\/\\dot{m}_\\mathrm{Edd}=4$, each followed by a $100\\ \\mathrm{Myr}$ phase of quiescence. The \\cite{T13} and our sample, could thus not contain any BHs that are actively accreting at the time of observation. \n\nSimulations suggest that the BH occupation fraction should be high enough for our field to contain high-redshift BHs. \\cite{Menou:2001aa} ran Monte Carlo simulations of the merger history of dark matter haloes. They showed that to reproduce the local BH distribution, $>3\\%$ of the $M_\\mathrm{halo}\\gtrsim10^8$ $M_{\\odot}$\\ haloes should be seeded with BHs at $z\\sim5$.  \n\n \\cite{Bellovary:2011aa} ran SPH+$N$-body simulations in which only the local gas properties, such as density, temperature and metallicity, influence the BH formation and evolution. They showed that the BH occupation fraction is halo mass dependent. At $z\\sim5$ they found a BH occupation fraction of $\\sim50\\%$ for $10^8$ $M_{\\odot}$\\ $<M_\\mathrm{halo}<10^{9}$ $M_{\\odot}$\\ haloes and a fraction of $100\\%$ for $M_\\mathrm{halo}>10^{9}$ $M_{\\odot}$\\ haloes. \n\n\\cite{Alexander:2014aa} presented a sophisticated model in which a BH seed is being fed by dense cold gas flows while it is embedded in a nuclear star cluster. This can lead to supra-exponential accretion and could explain how a light ($\\sim10$ $M_{\\odot}$) Pop III remnant  BH seed could grow into a $\\gtrsim10^4$ $M_{\\odot}$\\ seed within $\\sim10^7$ years. Nevertheless, these $\\gtrsim10^4$ $M_{\\odot}$\\ seeds still need to grow into the massive $10^9$ $M_{\\odot}$\\ sources that we find at $z>6$. This most likely happens via Eddington limited accretion. The distribution of high-redshift quasars that we observe at $z>6$ can be reproduced if the supra-exponential accretion and the subsequent Eddington-limited growth work efficiently in at least $1-5\\%$ of the dark matter haloes.     \n\n\\cite{Stark:2009aa}, \\cite{Vanzella:2009aa}, \\cite{Wilkins:2010aa}, \\cite{Bouwens:2014aa}, \\cite{Duncan:2014aa}  and many more have shown that the GOODS-south field contains hundreds of $z\\gtrsim5$ Lyman Break Galaxies. These high-redshift sources should have passed our manual inspection, the colour criteria, the stacking and the photometric redshift measurement. So, according to the simulations and the number of high-redshift Lyman Break Galaxies in our field, we would expect our sample to include high-redshift BHs. The number of actively accreting BHs could however still be low. For instance, it would be possible that only the most massive haloes host AGN.  \n\nFinding one or more high-redshift AGN would have opened up the window to the early BH growth era. \\cite{Treister:2011aa} showed that the massive and luminous quasars we observe at $z>6$ are most likely not representative of the entire high-redshift BH population. Such quasars are rare and presumably only constitute the high mass end of the entire BH population \\citep{Volonteri:2010aa}. At $z\\sim6$ we expect to find only $\\sim2$ in a 1000 $\\mathrm{deg}^{2}$ field \\citep{Fan:2001aa, Fan:2000aa}. Furthermore, these objects only allow us limited insight into seed formation scenarios. For typical seed masses $\\lesssim10^{6}$$M_{\\odot}$, these objects have to undergo multiple Salpeter times \\citep{Salpeter:1964aa} to reach $M_\\mathrm{BH}\\sim10^9$ $M_{\\odot}$. By the time we observe them as quasars, all initial seed information will be lost. We are hence especially interested in the population of more abundant, less massive, less luminous AGN that will end up in galaxies similar to the Milky Way. If our analysis had yielded a convincing high-redshift AGN candidate, we would have been able to put first constraints on this more representative and revealing BH population.         \n\nTo constrain the explanations for our results and to gain further insights into BH formation and growth at high redshift, this analysis needs to be repeated for a larger sample. Especially constraining the BH occupation fraction and the short, super-Eddington growth scenario requires a larger field. The new field does not need to be deeper, but wider than the $0.11\\ \\mathrm{deg}^{2}$ CDF-S \\citep{Luo:2008aa}. The 2.8-Ms \\textit{Chandra}\\ COSMOS Legacy Survey, for which the data will be available soon, covers a $2.2\\ \\mathrm{deg}^{2}$\\ area \\citep{Civano:2014aa}. Not being as deep as the 4-Ms \\textit{Chandra}\\ data, the \\textit{Chandra}\\ COSMOS Legacy Survey will probe a slightly different parameter space (see \\citealt{Treister:2011aa} for an illustration of the high-redshift number density that is necessary for an individual detection). Nonetheless, it will provide data for a much wider field and will thus enable us to repeat this analysis for a larger sample. ATHENA, which is meant to be launched in 2028, will allow us to constrain the BH occupation fraction to new accuracy. With its X-ray Wide Field Imager, ATHENA is meant to detect over 400 $z = 6 -8$ and over 30 $z > 8$ X-ray selected active BHs \\citep{Nandra:2013aa,Aird:2013aa}. JWST data could help to investigate the nature of our five low significance objects. \n\n\n\\section{Summary}\n\\label{sec:summary}\n\nWe searched for $z\\gtrsim5$ AGN in the \\textit{Chandra}\\ Deep Field South (CDF-S). We used the \\textit{Chandra}\\ 4-Ms catalog and combined it with GOODS\/ACS, CANDELS\/WFC3 and \\textit{Spitzer}\/IRAC data. Our main sample contained 58 sources. We ran a photometric redshift code, stacked the GOODS\/ACS data, applied colour criteria and the Lyman Break Technique. Furthermore, we used the X-ray Hardness Ratio as a redshift indicator. After combining all redshift tests, three final $z\\gtrsim5$ AGN candidates remained. Redetermining their photometric redshifts with additional VLT\/Hawk-I HUGS $K_S$-band and HST\/WFC3 HUDF $Y$-band data showed, that they are most likely low-redshift sources. We also found five sources that are detected in the X-rays, but that do not seem to possess a counterpart in the optical or infrared (low significance objects). The currently available data did not allow us to determine if these objects are possible high-redshift AGN candidates, spurious detections or optically faint low-redshift sources. Assuming that our three final candidates are indeed low-redshift sources and that our five low significance objects are either spurious detections or also at low redshift, we concluded that the CDF-S does not contain any high-redshift AGN. We also showed that we should have been able to detect active BHs in typical $z\\sim5$ Lyman Break Galaxies and why we would have expected to find at least some high $z$ AGN. \nOur results could be explained by:    \n\\begin{itemize}\n\\item a low BH occupation fraction or a low AGN fraction. If at high redshift only very few haloes contain a BH or only very few BHs are actively accreting, our sample could be too small to contain an AGN.   \n\\item BH growth via short, super-Eddington growth modes. If BHs primarily grow through short accretion episodes, the number of actively accreting BHs in our sample might be zero.\n\\item BH growth in optically faint galaxies. Our five low significance objects could indicate that high-redshift AGN primarily grow in galaxies that we do not detect in the optical. We were however unable to constrain their redshifts and noted that these could be spurious detections.  \n\\item BH growth via BH-BH mergers. If at high $z$ BHs primarily grow through mergers instead of accretion, X-rays might not trace BH growth.  \n\\end{itemize}\n\n\\begin{table*}\n\\tiny\n\t\\begin{center}\n\t\t\\begin{tabular}{lllllllllllllll}\n\t\t\\toprule\n\t\t\t{ } & {ID} & {$\\mathrm{RA}$} & {$\\mathrm{DEC}$} & {$\\mathrm{RA}_{H}$} & {$\\mathrm{DEC}_{H}$} & {visual} & {stacking} & {colour} & {Hardness} & {$\\sigma_{\\mathrm{HR}}$} & {$z_\\mathrm{phot}$} & {$\\sigma_{\\mathrm{phot},{+}}$} & {$\\sigma_{\\mathrm{phot},{-}}$} &{$\\chi^2$}\\\\\n\t\t\t{ } & { } & {(\\textit{Chandra})} & {(\\textit{Chandra})} & { } & { } & { } & { } & {criteria} & {Ratio (HR) } & { } & { } & { }\\\\ \n\t\t\t\\midrule\n\t\t\t$*$ & $121$ & $53.0268$ & $-27.7653$ & $53.0267$ & $-27.7653$ & $4$ & $4$ & $0$ & $0.21^{l}$ & $-$ & $1.26$ & $0.64$ & $0.04$ & $14.31$\\\\\n\t\t\t$*$ & $150$ & $53.0400$ & $-27.7985$ & $53.0398$ & $-27.7985$ & $4$ & $4$ & $0$ & $0.25^{l}$ & $-$ & $3.63$ & $0.14$ & $0.16$ & $1.66$\\\\\n\t\t\t$ $ & $173$ & $53.0477$ & $-27.8351$ & $53.0477$ & $-27.8351$ & $4$ & $4$ & $0$ & $-0.37$ & $0.13$ & $1.28$ & $0.30$ & $0.03$ & $4.11$\\\\\n\t\t\t$*$ & $184$ & $53.0523$ & $-27.7748$ & $53.0522$ & $-27.7747$ & $4$ & $4$ & $0$ & $0.19$ & $0.28$ & $1.73$ & $0.05$ & $0.05$ & $2.19$\\\\\n\t\t\t$*$ & $189$ & $53.0546$ & $-27.7931$ & $53.0544$ & $-27.7931$ & $4$ & $4$ & $0$ & $0.31^{l}$ & $-$ & $1.57$ & $0.17$ & $0.07$ & $9.34$\\\\\n\t\t\t$*$ & $199$ & $53.0579$ & $-27.8336$ & $53.0579$ & $-27.8336$ & $6$ & $7$ & $0$ & $-0.09$ & $0.10$ & $2.65$ & $0.11$ & $0.08$ & $16.61$\\\\\n\t\t\t$ $ & $211$ & $53.0620$ & $-27.8511$ & $53.0625$ & $-27.8508$ & $4$ & $4$ & $0$ & $0.06$ & $0.06$ & $1.38$ & $0.29$ & $0.07$ & $11.58$\\\\\n\t\t\t$*$ & $217$ & $53.0639$ & $-27.8438$ & $53.0638$ & $-27.8433$ & $0$ & $7$ & $0$ & $-0.17$ & $0.12$ & $3.72$ & $0.40$ & $0.23$ & $38.86$\\\\\n\t\t\t$ $ & $221$ & $53.0657$ & $-27.8790$ & $53.0660$ & $-27.8787$ & $4$ & $4$ & $0$ & $0.12^{l}$ & $-$ & $2.05$ & $0.09$ & $0.06$ & $8.84$\\\\\n\t\t\t$ $ & $226$ & $53.0668$ & $-27.8166$ & $53.0668$ & $-27.8165$ & $4$ & $4$ & $6$ & $0.33^{u}$ & $-$ & $5.43$ & $0.08$ & $0.12$ & $0.64$\\\\\n\t\t\t$*$ & $242$ & $53.0716$ & $-27.7699$ & $53.0713$ & $-27.7696$ & $4$ & $4$ & $0$ & $0.16$ & $0.26$ & $1.60$ & $0.06$ & $0.04$ & $16.75$\\\\\n\t\t\t$*$ & $244$ & $53.0721$ & $-27.8190$ & $53.0717$ & $-27.8187$ & $6$ & $7$ & $6$ & $0.29^{u}$ & $-$ & $1.90$ & $4.32$ & $0.22$ & $2.15$\\\\\n\t\t\t$ $ & $258$ & $53.0766$ & $-27.8641$ & $53.0766$ & $-27.8644$ & $6$ & $6$ & $0$ & $0.09^{l}$ & $-$ & $4.22$ & $0.62$ & $3.26$ & $12.49$\\\\\n\t\t\t$*$ & $273$ & $53.0821$ & $-27.7673$ & $53.0826$ & $-27.7681$ & $6$ & $7$ & $0$ & $0.03^{u}$ & $-$ & $3.45$ & $0.20$ & $0.83$ & $1.01$\\\\\n\t\t\t$ $ & $296$ & $53.0907$ & $-27.7825$ & $53.0913$ & $-27.7820$ & $6$ & $7$ & $6$ & $0.56^{l}$ & $-$ & $1.99$ & $0.10$ & $0.09$ & $12.71$\\\\\n\t\t\t$ $ & $301$ & $53.0924$ & $-27.8033$ & $53.0918$ & $-27.8028$ & $4$ & $4$ & $0$ & $0.34$ & $0.10$ & $2.02$ & $0.30$ & $0.29$ & $6.78$\\\\\n\t\t\t$*$ & $302$ & $53.0924$ & $-27.8268$ & $53.0923$ & $-27.8260$ & $4$ & $4$ & $0$ & $0.04^{u}$ & $-$ & $1.05$ & $0.75$ & $0.05$ & $26.22$\\\\\n\t\t\t$ $ & $303$ & $53.0925$ & $-27.8771$ & $53.0921$ & $-27.8767$ & $5$ & $0$ & $5$ & $0.35^{l}$ & $-$ & $0.73$ & $3.57$ & $-0.07$ & $0.37$\\\\\n\t\t\t$ $ & $306$ & $53.0939$ & $-27.8258$ & $53.0944$ & $-27.8259$ & $7$ & $7$ & $0$ & $0.16$ & $0.36$ & $2.87$ & $0.41$ & $0.11$ & $12.30$\\\\\n\t\t\t$*$ & $318$ & $53.0965$ & $-27.7449$ & $53.0966$ & $-27.7447$ & $4$ & $4$ & $0$ & $0.38^{u}$ & $-$ & $1.38$ & $0.06$ & $0.05$ & $34.03$\\\\\n\t\t\t$*$ & $321$ & $53.0984$ & $-27.7671$ & $53.0983$ & $-27.7667$ & $4$ & $4$ & $5$ & $0.31^{l}$ & $-$ & $1.22$ & $0.05$ & $0.03$ & $18.91$\\\\\n\t\t\t$ $ & $325$ & $53.1000$ & $-27.8086$ & $53.0995$ & $-27.8085$ & $4$ & $4$ & $0$ & $0.46^{u}$ & $-$ & $2.83$ & $0.23$ & $0.55$ & $7.27$\\\\\n\t\t\t$ $ & $328$ & $53.1016$ & $-27.8217$ & $53.1012$ & $-27.8224$ & $4$ & $4$ & $0$ & $0.24^{u}$ & $-$ & $1.68$ & $0.71$ & $1.06$ & $6.38$\\\\\n\t\t\t$ $ & $331$ & $53.1027$ & $-27.8606$ & $53.1028$ & $-27.8610$ & $4$ & $4$ & $0$ & $0.21^{u}$ & $-$ & $4.32$ & $0.02$ & $3.63$ & $3.33$\\\\\n\t\t\t$*$ & $348$ & $53.1052$ & $-27.8752$ & $53.1058$ & $-27.8753$ & $6$ & $6$ & $0$ & $0.15^{u}$ & $-$ & $3.11$ & $0.12$ & $0.67$ & $3.85$\\\\\n\t\t\t$*$ & $354$ & $53.1076$ & $-27.8558$ & $53.1079$ & $-27.8558$ & $7$ & $7$ & $0$ & $-0.54$ & $0.15$ & $2.83$ & $3.58$ & $0.85$ & $0.38$\\\\\n\t\t\t$ $ & $371$ & $53.1116$ & $-27.7679$ & $53.1118$ & $-27.7680$ & $5$ & $5$ & $0$ & $0.15$ & $0.10$ & $5.56$ & $0.04$ & $4.50$ & $0.61$\\\\\n\t\t\t$ $ & $373$ & $53.1118$ & $-27.9096$ & $53.1113$ & $-27.9094$ & $4$ & $4$ & $4$ & $0.22^{u}$ & $-$ & $2.54$ & $0.15$ & $0.25$ & $12.99$\\\\\n\t\t\t$*$ & $389$ & $53.1193$ & $-27.7659$ & $53.1186$ & $-27.7658$ & $4$ & $4$ & $0$ & $0.16^{u}$ & $-$ & $2.02$ & $0.05$ & $0.48$ & $7.31$\\\\\n\t\t\t$*$ & $392$ & $53.1199$ & $-27.7432$ & $53.1198$ & $-27.7436$ & $7$ & $5$ & $0$ & $0.05^{u}$ & $-$ & $3.07$ & $3.10$ & $0.00$ & $1.31$\\\\\n\t\t\t$*$ & $402$ & $53.1219$ & $-27.7529$ & $53.1214$ & $-27.7531$ & $5$ & $5$ & $0$ & $0.27^{l}$ & $-$ & $2.37$ & $0.64$ & $0.21$ & $2.91$\\\\\n\t\t\t$ $ & $403$ & $53.1220$ & $-27.9388$ & $53.1224$ & $-27.9381$ & $4$ & $4$ & $0$ & $0.20^{l}$ & $-$ & $4.22$ & $0.17$ & $0.12$ & $6.97$\\\\\n\t\t\t$ $ & $410$ & $53.1241$ & $-27.8913$ & $53.1242$ & $-27.8917$ & $4$ & $4$ & $0$ & $0.54$ & $0.12$ & $3.49$ & $0.27$ & $0.27$ & $2.63$\\\\\n\t\t\t$*$ & $428$ & $53.1296$ & $-27.8278$ & $53.1295$ & $-27.8276$ & $4$ & $4$ & $0$ & $0.24^{u}$ & $-$ & $1.15$ & $0.07$ & $0.04$ & $17.71$\\\\\n\t\t\t$ $ & $430$ & $53.1305$ & $-27.7912$ & $53.1310$ & $-27.7911$ & $0$ & $7$ & $0$ & $0.29$ & $0.15$ & $6.54$ & $1.34$ & $5.07$ & $0.03$\\\\\t\t\t\n\t\t\t$*$ & $444$ & $53.1340$ & $-27.7811$ & $53.1340$ & $-27.7809$ & $4$ & $4$ & $4$ & $0.28$ & $0.07$ & $1.42$ & $0.10$ & $0.06$ & $41.98$\\\\\n\t\t\t$*$ & $455$ & $53.1378$ & $-27.8022$ & $53.1378$ & $-27.8021$ & $4$ & $4$ & $0$ & $0.21^{u}$ & $-$ & $1.26$ & $0.05$ & $0.03$ & $5.63$\\\\\n\t\t\t$ $ & $456$ & $53.1380$ & $-27.8683$ & $53.1381$ & $-27.8684$ & $7$ & $7$ & $0$ & $-0.29$ & $0.11$ & $10.91$ & $-0.92$ & $8.39$ & $0.00$\\\\\n\t\t\t$*$ & $460$ & $53.1393$ & $-27.8745$ & $53.1394$ & $-27.8746$ & $0$ & $5$ & $0$ & $-0.02$ & $0.27$ & $2.91$ & $2.68$ & $0.09$ & $0.73$\\\\\n\t\t\t$*$ & $462$ & $53.1403$ & $-27.7976$ & $53.1405$ & $-27.7973$ & $4$ & $4$ & $0$ & $0.06$ & $0.39$ & $1.63$ & $0.06$ & $0.04$ & $5.12$\\\\\n\t\t\t$ $ & $466$ & $53.1417$ & $-27.8167$ & $53.1416$ & $-27.8166$ & $4$ & $4$ & $0$ & $0.27$ & $0.13$ & $2.02$ & $0.20$ & $0.31$ & $5.73$\\\\\n\t\t\t$*$ & $485$ & $53.1466$ & $-27.8710$ & $53.1460$ & $-27.8711$ & $7$ & $7$ & $0$ & $0.45$ & $0.15$ & $2.83$ & $0.54$ & $0.68$ & $14.74$\\\\\n\t\t\t$ $ & $496$ & $53.1505$ & $-27.8890$ & $53.1507$ & $-27.8886$ & $4$ & $4$ & $0$ & $0.00$ & $-$ & $2.50$ & $0.31$ & $2.11$ & $1.44$\\\\\n\t\t\t$*$ & $522$ & $53.1585$ & $-27.7741$ & $53.1583$ & $-27.7738$ & $4$ & $5$ & $6$ & $-0.40$ & $0.04$ & $1.60$ & $0.21$ & $0.05$ & $4.06$\\\\\n\t\t\t$ $ & $535$ & $53.1627$ & $-27.7443$ & $53.1622$ & $-27.7442$ & $4$ & $4$ & $0$ & $-0.12$ & $0.06$ & $2.37$ & $0.30$ & $0.39$ & $3.44$\\\\\n\t\t\t$*$ & $539$ & $53.1632$ & $-27.8091$ & $53.1621$ & $-27.8097$ & $4$ & $4$ & $0$ & $0.53^{l}$ & $-$ & $0.97$ & $0.06$ & $0.08$ & $44.03$\\\\\n\t\t\t$ $ & $546$ & $53.1653$ & $-27.8142$ & $53.1648$ & $-27.8144$ & $4$ & $4$ & $0$ & $0.29$ & $0.03$ & $3.15$ & $0.18$ & $0.25$ & $8.09$\\\\\n\t\t\t$ $ & $556$ & $53.1701$ & $-27.9298$ & $53.1699$ & $-27.9304$ & $4$ & $4$ & $0$ & $0.40$ & $0.03$ & $2.68$ & $0.17$ & $0.16$ & $12.11$\\\\\n\t\t\t$*$ & $574$ & $53.1787$ & $-27.8027$ & $53.1782$ & $-27.8027$ & $6$ & $7$ & $0$ & $0.38$ & $0.25$ & $3.32$ & $0.42$ & $0.75$ & $2.72$\\\\\n\t\t\t$ $ & $578$ & $53.1806$ & $-27.7797$ & $53.1807$ & $-27.7796$ & $7$ & $7$ & $0$ & $0.00$ & $-$ & $6.77$ & $0.97$ & $5.10$ & $0.00$\\\\\n\t\t\t$ $ & $583$ & $53.1835$ & $-27.7766$ & $53.1834$ & $-27.7764$ & $7$ & $7$ & $0$ & $-0.51$ & $0.06$ & $9.36$ & $0.63$ & $4.82$ & $0.00$\\\\\n\t\t\t$ $ & $589$ & $53.1850$ & $-27.8198$ & $53.1851$ & $-27.8196$ & $4$ & $4$ & $6$ & $0.17^{u}$ & $-$ & $1.28$ & $0.26$ & $0.08$ & $5.77$\\\\\n\t\t\t$*$ & $591$ & $53.1852$ & $-27.7174$ & $53.1848$ & $-27.7173$ & $4$ & $4$ & $0$ & $0.29^{u}$ & $-$ & $1.35$ & $0.17$ & $0.21$ & $21.54$\\\\\n\t\t\t$ $ & $620$ & $53.1960$ & $-27.8927$ & $53.1957$ & $-27.8928$ & $4$ & $4$ & $0$ & $-0.02$ & $0.06$ & $2.57$ & $0.55$ & $0.45$ & $4.34$\\\\\n\t\t\t$ $ & $624$ & $53.1981$ & $-27.8323$ & $53.1979$ & $-27.8319$ & $5$ & $6$ & $0$ & $0.33^{l}$ & $-$ & $1.52$ & $2.23$ & $0.07$ & $0.56$\\\\\n\t\t\t$*$ & $625$ & $53.1989$ & $-27.8440$ & $53.1989$ & $-27.8439$ & $4$ & $4$ & $0$ & $0.71$ & $0.10$ & $1.68$ & $0.07$ & $0.04$ & $16.22$\\\\\n\t\t\t$*$ & $630$ & $53.2016$ & $-27.8443$ & $53.2027$ & $-27.8448$ & $7$ & $4$ & $0$ & $0.36^{u}$ & $-$ & $2.05$ & $5.04$ & $0.16$ & $0.37$\\\\\n\t\t\t$*$ & $651$ & $53.2153$ & $-27.8703$ & $53.2150$ & $-27.8695$ & $7$ & $0$ & $0$ & $0.40$ & $0.10$ & $1.90$ & $0.41$ & $0.34$ & $16.22$\\\\\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\end{center}\n\t\\caption{\\label{tab:overview}Overview over the redshift estimates that we gained for each object in the course of this analysis. All photometric redshifts were gained by using GOODS\/ACS, CANDELS and in some cases also \\textit{Spitzer}\\ data (marked with an asterisk). For our three final candidates (456, 578, 583) we also computed photometric redshifts using HUDF and HUGS data. Please see sections \\ref{sec:HUDF} and \\ref{sec:HUGS} for the corresponding values.  Note that the stacking procedure only gives a redshift indication. Hence, if '4' is given as the stacking result this corresponds to $z\\lesssim4$,  '7' indicates $z\\gtrsim7$. We mark sources for which we only gained upper or lower limits on the Hardness Ratio with '$u$' and '$l$'. '0' indicates that a source could not be classified by the corresponding redshift test.}\t\n\\end{table*}\n\\newpage\n\n\\section*{Acknowledgements}\nWe thank  Andreas Faisst for helpful discussions and Richard Ellis for suggesting the stacking analysis. We also thank the anonymous referee for helpful comments. AKW, KS and MK gratefully acknowledge support from Swiss National Science Foundation Grant PP00P2\\_138979\/1. Support for the work of ET was provided by the Center of Excellence in Astrophysics and Associated Technologies (PFB 06), by the FONDECYT regular grant 1120061 and by the CONICYT Anillo project ACT1101. This research has made use of NASA's ADS Service. This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013).\n\\newpage\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n    \n    \\para{Background}\n    The problem of simulating the time evolution of a quantum system is perhaps the most important application of quantum computers. Indeed, this was the reason Feynman proposed quantum computing~\\cite{Fey82}, and it remains an important practical application since a significant fraction of the world's supercomputing power is used to solve instances of this problem that arise in materials science, condensed matter physics, high energy physics, and chemistry~\\cite{NERSC16}.\n    \n    All known classical algorithms (i.e., algorithms that run on traditional non-quantum computers) for this problem run in exponential time. On the other hand, from the early days of quantum computing~\\cite{Fey82,Lloyd1996} it was known that quantum computers can solve this problem in polynomial time. More precisely, when formalized as a decision problem, the problem of simulating the time evolution of a quantum system is in the complexity class $\\mathsf{BQP}$, the class of problems solved by a quantum computer to bounded error in polynomial time. Furthermore, the problem is complete for $\\mathsf{BQP}$~\\cite{Feynman1985,Nagaj2009},\n    which means we do not expect there to be efficient classical algorithms for the problem, since that would imply $\\mathsf{BPP}=\\mathsf{BQP}$, which in turn would imply polynomial-time algorithms for problems such as integer factorization and discrete log~\\cite{Sho97}. \n    \n   \n    \n    \\para{Hamiltonian simulation problem} The Hamiltonian simulation problem is a standard formalization of the problem of simulating the time evolution\\footnote{This is sometimes referred to as ``real time evolution'', to distinguish it from ``imaginary time evolution'' which we will not talk about in this paper.} of a quantum system. In this problem, we assume the quantum system whose time evolution we wish to simulate consists of $n$ qubits and we want to simulate its time evolution for time $T$, in the sense that we are provided with the initial state $|\\psi(0)\\>$ and we want to compute the state of the system at time $T$, $|\\psi(T)\\>$. The goal of an efficient simulation is to solve the problem in time polynomial in $n$ and $T$.\n    \n    The state of a system of $n$ qubits can be described by a complex vector of dimension $2^n$ of unit norm. Since we are studying quantum algorithms for the problem, we are given the input as an $n$-qubit quantum state, and have to output an $n$-qubit quantum state. The relation between the output state at time $T$ and the initial state at time $0$ is given by the Schr\\\"odinger equation\n    \\begin{equation}\n    i\\frac{\\mathrm{d}}{\\mathrm{d}t}|\\psi(t)\\>=H(t)|\\psi(t)\\>,\n    \\end{equation}\n    where the Hamiltonian $H$, a $2^n \\times 2^n$ complex Hermitian matrix, has entries which may also be functions of time. The Hamiltonian captures the interaction between the constituents of the system and governs time dynamics. In the special case where the Hamiltonian is independent of time, the  Schr\\\"odinger equation can be solved to yield  $|\\psi(T)\\>=e^{-iHT}|\\psi(0)\\>$.\n    \n    More formally, the input to the Hamiltonian simulation problem consists of a Hamiltonian $H$ (or $H(t)$ in the time-dependent case), a time $T$, and an error parameter $\\epsilon$. The goal is to output a quantum circuit that approximates the unitary matrix that performs the time evolution above (e.g., for time-independent Hamiltonians, the quantum circuit should approximate the unitary $e^{-iHT}$). The notion of approximation used is the spectral norm distance between the ideal unitary and the one performed by the circuit. \n    The cost of a quantum circuit is measured by the number of gates used in the circuit, where the gates come from some fixed universal gate set. Note that it is important to describe how the Hamiltonian in the input is specified, since it is a matrix of size $2^n \\times 2^n$. This will be made precise when talking about specific classes of Hamiltonians that we would like to simulate.\n    \n    \\para{Geometrically local Hamiltonian simulation}\n    The most general class of Hamiltonians that is commonly studied in the literature is the class of sparse Hamiltonians~\\cite{AT03,Chi04,BAC+07,BerryEtAl2014,TS,BCK15,QSP,LC16, LC17USA}. A Hamiltonian on $n$ qubits is sparse if it has only $\\poly(n)$ nonzero entries in any row or column. For such Hamiltonians, we assume we have an efficient algorithm to compute the nonzero entries in each row or column, and the input Hamiltonian is specified by an oracle that can be queried for this information. In this model, recent quantum algorithms have achieved optimal complexity in terms of the queries made to the oracle~\\cite{QSP}.\n    \n    A very important special case of this type of Hamiltonian is a ``local Hamiltonian.'' Confusingly, this term is used to describe two different kinds of Hamiltonians in the literature. \n    We distinguish these two definitions of ``local'' by referring to them as ``non-geometrically local'' and ``geometrically local'' in this introduction. A non-geometrically local Hamiltonian is a Hamiltonian $H(t)$ that can be written as a sum of polynomially many terms $H_j(t)$, each of which acts nontrivially on only $k$ qubits at a time (i.e., the matrix acts as identity on all other qubits). A geometrically local Hamiltonian is similar, except that each term $H_j(t)$ must act on $k$ adjacent qubits. Since we refer to ``adjacent'' qubits, the geometry of how the qubits are laid out in space must be specified. In this paper we will deal with qubits laid out in a $D$-dimensional lattice in Euclidean space. I.e., qubits are located at points in $\\mathbb{Z}^D$ and are adjacent if they are close in Euclidean distance.\n    \n    Lattice Hamiltonians in $D$-dimensions (with $D\\leq 3$) already model all the physical systems we are interested in,\n    are of fundamental importance by the principle of locality.\n    From a practical perspective, this case captures a large fraction of all physical systems we are interested in.\\footnote{There are some physical situations where we do care about more general Hamiltonians. Even though the system we are given may be described by a geometrically local Hamiltonian, it is sometimes computationally advantageous to represent a given system with a non-geometrically local (or sparse) Hamiltonian.}\n    \n   \n    \n    \\para{Prior best algorithms}\n    To describe the known algorithms for this problem, we need to formally specify the problem. Although our results apply to very general time-dependent Hamiltonians, while comparing to previous work we assume the simpler case where the Hamiltonian is time independent.\n    \n    We assume our $n$ qubits are laid out in a $D$-dimensional lattice $\\Lambda$ in $\\mathbb{R}^D$, where $D=\\O(1)$, and every unit ball contains $\\O(1)$ qubits. We assume our Hamiltonian $H$ is given as a sum of terms $H = \\sum_{X\\subseteq\\Lambda} h_X$, where each $h_X$ only acts nontrivally on qubits in $X$ (and acts as identity on the qubits in $\\Lambda \\setminus X$), such that $h_X=0$ if $\\diam(X)>1$,\n    which enforces geometric locality. \n    (More formally, we rescale the metric\n    in such a way that $\\diam(X) > 1$ implies $h_X = 0$.)\n    We normalize the Hamiltonian by requiring $\\norm{h_X}\\leq 1$.\n    \n    We consider a quantum circuit simulating the time evolution due to such a Hamiltonian efficient if it uses $\\poly(n,T,1\/\\epsilon)$ gates. To get some intuition for what we should hope for, notice that in the real world, time evolution takes time $T$ and uses $n$ qubits. Regarding ``Nature'' as a quantum simulator, we might expect that there is a quantum circuit that uses $O(n)$ qubits, $O(T)$ circuit depth, and $O(nT)$ total gates to solve the problem. It is also reasonable to allow logarithmic overhead in the simulation since such overheads are common even when one classical system simulates the time evolution of another (e.g., when one kind of Turing machine simulates another).\n    \n    However, previous algorithms for this problem fall short of this expectation. The best Hamiltonian simulation algorithms for sparse Hamiltonians~\\cite{TS,QSP,LC16} have query complexity $\\O(nT \\polylog(nT\/\\epsilon))$, but the assumed oracle for the entries requires $O(n)$ gates to implement, yielding an algorithm that uses $\\O(n^2T \\polylog(nT\/\\epsilon))$ gates. This was also observed in a recent paper of Childs, Maslov, Nam, Ross, and Su~\\cite{Childs2017}, who noted that for $T=n$, all the sparse Hamiltonian simulation algorithms had gate cost proportional to $n^3$ (or worse). A standard application of high-order Lie-Trotter-Suzuki expansions~\\cite{Trotter1959,Suzuki1991,Lloyd1996,BAC+07} yields gate complexity $O(n^2T (nT\/\\epsilon)^{\\delta})$ for any fixed $\\delta > 0$. It has been argued~\\cite[Sec.~4.3]{JordanLeePreskill2014} that \n    this in fact yields an algorithm with gate complexity $O(nT (nT\/\\epsilon)^{\\delta})$ for any fixed $\\delta > 0$. We believe this analysis is correct, but perhaps some details need to be filled in to make the analysis rigorous. In any case, this algorithm still performs  worse than desired, and in particular does not have polylogarithmic dependence on $1\/\\epsilon$.\n    \n    \\subsection{Results} We exhibit a quantum algorithm that simulates the time evolution due to a time-dependent lattice Hamiltonian with a circuit that uses $\\O(nT \\polylog(nT\/\\epsilon))$ geometrically local $2$-qubit gates (i.e., the gates in our circuit also respect the geometry of the qubits), with depth  $\\O(T \\polylog(nT\/\\epsilon))$ using only $\\polylog(nT\/\\epsilon)$ ancilla qubits. We then also prove a matching lower bound, showing that no quantum algorithm can do better (up to logarithmic factors), even if we relax the output requirement significantly. We now describe our results more formally.\n    \n    \\para{Algorithmic results}\n    We consider a more general version of the problem with time-dependent Hamiltonians. In this case we will have $H(t) = \\sum_{X\\subseteq \\Lambda} h_X(t)$ with the locality and norm conditions as before. However, now the operators $h_X(t)$ are functions of time and we need to impose some reasonable constraints on the entries to obtain polynomial-time algorithms.\n    \n    First we need to be able to compute the entries of our Hamiltonian efficiently at a given time $t$.\n    We say that a function $\\alpha: [0,T] \\ni t \\mapsto \\alpha(t) \\in  {\\mathbb{R}}$ \n    is \\emph{efficiently computable}\n    if there is an algorithm that outputs $\\alpha(t)$ to precision $\\epsilon$ \n    for any given input $t$ specified to precision $\\epsilon$\n    in running time $\\polylog(T\/\\epsilon)$.\n    Note that any complex-valued analytic function \n    on a nonzero neighborhood of a closed real interval in the complex plane\n    is efficiently computable (see \\cref{app:chebyshev}).\n    We will assume that each entry in a local term $h_X(t)$ is efficiently computable.\n    \n    In addition to being able to compute the entries of the Hamiltonian, we require that the entries do not change wildly with time; otherwise, a sample of entries at discrete times may not predict the behavior of the entries at other times.\n    We say that a function $\\alpha$ on the interval $[0,T]$ ($T \\ge 1$) is \n    \\emph{piecewise slowly varying}\n    if there are $M = {{O}}(T)$ intervals $[t_{j-1},t_j]$ \n    with $0 = t_0 < t_1 < \\cdots < t_M = T$\n    such that $\\frac{\\mathrm{d}}{\\mathrm{d}t}\\alpha(t)$ exists and is bounded \n    by $1\/(t_j - t_{j-1})$ for $t \\in (t_{j-1},t_j)$.\n    In particular, a function is piecewise slowly varying if it is a sum of $O(T)$ pieces, each of which has derivative at most $O(1)$. \n    We will assume that each entry in a term $h_X(t)$ is piecewise slowly varying. \n    \n    We are now ready to state our main result, which is proved in \\Cref{sec:algorithm}\n    \n    \\begin{restatable}{theorem}{main}\n        Let $H(t) = \\sum_{X \\subseteq \\Lambda} h_X(t)$ be a time-dependent Hamiltonian\n        on a lattice $\\Lambda$ of $n$ qubits, embedded in the Euclidean metric space $\\mathbb R^D$.\n        Assume that every unit ball contains ${{O}}(1)$ qubits and\t$h_X = 0$ if $\\diam(X) > 1$.\n        Also assume that every local term $h_X(t)$ is efficiently computable (e.g., analytic), piecewise slowly varying on time domain $[0,T]$, and has  $\\norm{h_X(t)} \\le 1$ for any $X$ and $t$.\n        Then, there exists a quantum algorithm that can approximate the time evolution of $H$\n        for time $T$ to accuracy $\\epsilon$ using ${{O}}( Tn \\polylog(Tn\/\\epsilon))$ 2-qubit local gates,\n        and has depth ${{O}}(T \\polylog(Tn\/\\epsilon))$.\n        \\label{thm:main}\n    \\end{restatable}\n    \n    Our algorithm uses $O(1)$ ancillas per system qubit on which $H$ is defined.\n    The ancillas are interspersed with the system qubits,\n    and all the gates respect the locality of the lattice.\n    \n    \\para{Lower bounds}\n    We also prove a lower bound on the gate complexity of problem of simulating the time evolution of a time-dependent lattice Hamiltonian. This lower bound matches, up to logarithmic factors, the gate complexity of the algorithm presented in \\Cref{thm:main}. Note that unlike previous lower bounds on Hamiltonian simulation~\\cite{BAC+07,BerryEtAl2014,BCK15}, which prove lower bounds on query complexity, this is a lower bound on the number of gates required to approximately implement the time-evolution unitary. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem. For concreteness, we focus on a 1-dimensional time-dependent local Hamiltonian in this section, although the lower bound extends to other constant dimensions with minor modifications. The lower bounds are proved in \\Cref{sec:lowerbound}.\n    \n    Before stating the result formally, let us precisely define the class of Hamiltonians for which we prove the lower bound.\n    We say a Hamiltonian $H(t)$ acting on $n$ qubits is a \n    ``piecewise constant 1D Hamiltonian''\n    if $H(t)=\\sum_{j=1}^{n-1}H_j(t)$, \n    where $H_j(t)$ is only supported on qubits $j$ and $j+1$ with $\\max_t \\norm{H_j(t)} = {{O}}(1)$,\n    and there is a time slicing $0 = t_0 < t_1 < \\cdots < t_M = T$\n    where $t_m - t_{m-1} \\le 1$ and $M = {{O}}(T)$ such that \n    $H(t)$ is time-independent within each time slice.\n    \n    For such Hamiltonians, the time evolution operator for time $T$ can be simulated with error at most $\\epsilon$ \n    using \\Cref{thm:main} with ${{O}}(Tn \\polylog(Tn\/\\epsilon))$ $2$-qubit local gates \n    (i.e., the 2-qubit gates only act on adjacent qubits). \n    In particular, for any constant error, the simulation only requires ${\\widetilde{\\calO}}(Tn)$ 2-qubit local gates. \n    We prove a matching lower bound, where the lower bound even holds against circuits that may use non-geometrically local (i.e., acting on non-adjacent qubits) $2$-qubit gates from a possibly infinite gate set and unlimited ancilla qubits. \n    \n    \\begin{restatable}{theorem}{lowerboundgeneral}\n        \\label{thm:lowerboundgeneral}\n        For any integers $n$ and $T \\leq 4^n$, \n        there exists a piecewise constant bounded 1D Hamil{-}tonian $H(t)$ on $n$ qubits,\n        such that any quantum circuit that approximates the time evolution due to $H(t)$ for time $T$ to constant error must use ${\\widetilde{\\Omega}}(Tn)$ $2$-qubit gates. \n        The quantum circuit may use unlimited ancilla qubits and the gates may be non-local and come from a possibly infinite gate set.\n    \\end{restatable}\n    \n    \n    Note that this lower bound only holds for $T\\leq 4^n$, because any unitary on $n$ qubits can be implemented with ${\\widetilde{\\calO}}(4^n)$ 2-qubit local gates~\\cite{BBC+95,Kni95}. \n    \n    We can also strengthen our lower bound to work in the situation where we are only interested in measuring a local observable at the end of the simulation. The simulation algorithm presented in \\Cref{thm:main} provides a strong guarantee: the output state is $\\epsilon$-close to the ideal output state in trace distance. Trace distance captures distinguishability with respect to arbitrary measurements, but for some applications it might be sufficient for the output state to be close to the ideal state with respect to local measurements only. We show that even in this limited measurement setting, it is not possible to speed up our algorithm in general. In fact, our lower bound works even if the only local measurement performed is a computational basis measurement on the first output qubit.\n    \n    \\begin{restatable}{theorem}{lowerboundlocal}\n        \\label{thm:lowerboundlocal}\n        For any integers $n$ and $T$ such that $1 \\le n\\leq T\\leq 2^n$, \n        there exists a piecewise constant bounded 1D Hamiltonian $H(t)$ on $n$ qubits, \n        such that any quantum circuit that approximates the time evolution due to $H(t)$ for time $T$ to constant error on any local observable must use ${\\widetilde{\\Omega}}(Tn)$ 2-qubit gates. \n        If $T\\leq n$, we have a lower bound of ${\\widetilde{\\Omega}}(T^2)$ gates. \n        (The quantum circuit may use unlimited ancilla qubits and the gates may be non-local and come from a possibly infinite gate set.)\n    \\end{restatable}\n    \n    Note that the fact that we get a weaker lower bound of ${\\widetilde{\\Omega}}(T^2)$ when $T\\leq n$ is not a limitation, but reflects the fact that small time evolutions are actually easier to simulate when the measurement is local. To see this, consider first simulating the time evolution using the algorithm in \\Cref{thm:main}. This yields a circuit with ${\\widetilde{\\calO}}(Tn)$ 2-qubit local gates. But if we only want the output of a local measurement after time $T$, qubits that are far away from the measured qubits cannot affect the output, since the circuit only consists of 2-qubit local gates. Hence we can simply remove all gates that are more than distance equal to the depth of the circuit, ${\\widetilde{\\calO}}(T)$, away from the measured qubits. We are then left with a circuit that uses ${\\widetilde{\\calO}}(T^2)$ gates, matching the lower bound in \\Cref{thm:lowerboundlocal}.\n    \n\\subsection{Techniques}\n    \n    \\para{Algorithm}\n    Our algorithm is based on a decomposition of the time evolution unitary using Lieb-Robinson bounds~\\cite{LiebRobinson1972,Hastings2004LSM,NachtergaeleSims2006,HastingsKoma2006,Hastings2010},\n    that was made explicit by Osborne~\\cite{Osborne2006} (see also Michalakis~\\cite[Sec.~III]{Michalakis2012}),\n    which when combined with recent advances in Hamiltonian simulation~\\cite{TS,QSP,LC16},\n    yields \\Cref{thm:main}. \n   \n   \n    \n    Lieb-Robinson bounds are theorems that informally state that information travels at a constant speed in geometrically local Hamiltonians. For intuition, consider a 1-dimensional lattice of qubits and a geometrically local Hamiltonian that is evolved for a short amount of time. If the time is too short, no information about the first qubit can be transmitted to the last qubit. Lieb-Robinson bounds make this intuition precise, and show that the qubit at position $n$ is only affected by the qubits and operators at position 1 after time $\\Omega(n)$. Note that if this were a small-depth unitary circuit of geometrically local $2$-qubit gates such a statement would follow using a ``lightcone'' argument. In other words, after one layer of geometrically local $2$-qubit gates, the influence of qubit 1 can only have spread to qubit 2. Similarly, after $k$ layers of $2$-qubit gates, the influence of qubit 1 can only have spread up to qubit $k$. The fact that this extends to geometrically local Hamiltonians is nontrivial, and is only approximately true. See \\Cref{lem:LR} for a formal statement of a Lieb-Robinson bound.\n    \n    We use these ideas to chop up the large unitary that performs time evolution for the full Hamiltonian $H$ into many smaller unitaries that perform time evolution for a small portion of the Hamiltonian. Quantitatively, we break Hamiltonian simulation for $H$ for time $O(1)$ into $O(n\/\\log(nT\/\\epsilon))$ pieces, each of which is a Hamiltonian simulation problem for a Hamiltonian on an instance of size $O(\\log(nT\/\\epsilon))$ to exponentially small error. At this point we can use any Hamiltonian simulation algorithm for the smaller piece as long as it has polynomial gate cost and has exponentially good dependence on $\\epsilon$. While Hamiltonian simulation algorithms based on product formulas do not have error dependence that is $\\polylog(1\/\\epsilon)$, recent Hamiltonian simulation algorithms, such as \\cite{BerryEtAl2014,TS,BCK15,QSP,LC16} have $\\polylog(1\/\\epsilon)$ scaling. Thus our result crucially uses the recent advances in Hamiltonian simulation with improved error scaling.\n    \n    \\para{Lower bound} As noted before, we lower bound the gate complexity (or total number of gates) required for Hamiltonian simulation, which is different from prior work which proved lower bounds on the query complexity of Hamiltonian simulation. As such, our techniques are completely different from those used in prior work. Informally, our lower bounds are based on a refined circuit-size hierarchy theorem for quantum circuits, although we are technically comparing two different resources in two different models, which are simulation time for Hamiltonians versus gate cost for circuits. \n    \n    As a simple motivating example, consider circuit-size hierarchy theorems for classical or quantum circuits more generally. Abstractly, a hierarchy theorem generally states that a computational model with $X$ amount of a resource (e.g., time, space, gates) can do more if given more of the same resource. For example, it can be shown that for every $G \\ll 2^n\/n$, there exists a Boolean function on $n$ bits that cannot be computed by a circuit of size $G$, but can be computed by a circuit of size $G+O(n)$. We show similar hierarchy theorems for quantum circuit size, except that we show that the circuit of larger size that computes the function actually comes from a weaker family of circuits. Informally, we are able to show that there are functions that can be computed by a larger circuit that uses only geometrically local $2$-qubit gates from a fixed universal gate set that cannot be a computed by a smaller circuit, even if we allow the smaller circuit access to unlimited ancilla bits and non-geometrically local $2$-qubit from an infinite gate set. We then leverage this asymmetric circuit size hierarchy theorem to show that there is a Hamiltonian whose evolution for time $T$ cannot be simulated by a circuit of size $\\ll nT$, by embedding the result of any quantum circuit with geometrically local $2$-qubit gates into a piecewise constant Hamiltonian with time proportional to the depth of the circuit.\n    \n    \n    \\section{Algorithm and analysis}\n    \\label{sec:algorithm}\n    \n    In this section we establish our main algorithmic result, restated below for convenience:\n    \n    \\begin{figure*}[t]\n        \\centering\n        \\includegraphics[width=0.85\\textwidth]{algorithm.pdf}\n        \\caption{Decomposition of time evolution operator for time $t = {{O}}(1)$.\n            The time is going upwards.\n            Each block $\\square$ represents the forward time evolution, $e^{-it H_\\square}$,\n            if the arrow is upward,\n            and backward time evolution, $e^{+it H_\\square}$, if the arrow is downward.\n            Here, $H_\\square$ is the sum of local terms in the Hamiltonian \n            supported within the block.\n            The overlap has size $\\ell$.\n            (a) shows a one-dimensional setting, \n            but a generalization to higher $D$ dimensions is readily achieved\n            by regarding each block as a $(D-1)$-dimensional hyperplane\n            so that the problem reduces to lower dimensions.\n            (b) shows a two-dimensional setting.\n            The approximation error from the depicted decomposition is $\\epsilon = {{O}}(e^{-\\mu \\ell} L^D \/ \\ell)$\n            where $L$ is the linear system size, $\\ell$ is the width of the overlap between blocks,\n            and $\\mu > 0$ is a constant that depends only on the locality of the Hamiltonian.\n            One can use any algorithm to further \n            decompose the resulting ``small'' unitaries on ${{O}}(\\log(L\/\\epsilon))$ qubits\n            into elementary gates.\n            To achieve gate count that is linear (up to logarithmic factors) in spacetime volume,\n            the algorithm for simulating the blocks needs to be polynomial in the block size and polylogarithmic in accuracy.\n        }\n        \\label{fig:algorithm}\n    \\end{figure*}\n    \n    \n    \n    \\main*\n    \n    The algorithm is depicted in \\Cref{fig:algorithm}.\n    Before showing why this algorithm works, we provide a high-level overview of the algorithm\n    and the structure of the proof.\n    Since a time evolution unitary $U(T;0)$ is equal to \n    $U(T=t_M; t_{M-1})U(t_{M-1},t_{M-2}) \\cdots U(t_2;t_1)U(t_1; t_0 = 0)$,\n    we will focus on a time evolution operator $U(t;0)$ where $t = {{O}}(1)$,\n    generated by a slowly varying bounded Hamiltonian.\n    The key idea, as shown in \\Cref{fig:algorithm}, is that the time-evolution operator, \n    $e^{-itH}$ due to the full Hamiltonian $\\sum_{X\\subseteq \\Lambda} h_X$ \n    can be approximately written as a product \n    \\begin{align}\n    e^{-itH} \\approx \n    \\left( e^{-itH_A} \\right) \\left(   e^{+itH_Y}  \\right) \\left(  e^{-itH_{Y\\cup B}} \\right).\n    \\end{align}\n    Here $A\\cup B = \\Lambda$ and \n    we think of $A$ and $B$ as large regions, but $Y$ as a small subset of $A$.\n    The error in the approximation is exponentially small as long as $Y$ is large enough.\n    This is formally proved in \\Cref{lem:decomposition}, which is supported by \n    \\Cref{lem:ode} and \\Cref{lem:LRB}. Applying this twice, using \n    \\begin{align}\n    e^{-itH_{Y\\cup B}} \n    \\approx \n    \\left( e^{-itH_B} \\right)\n    \\left( e^{+itH_Z}  \\right)\n    \\left( e^{-itH_{Y \\cup Z}} \\right)\n    \\end{align}\n    leads to a symmetric approximation\n    as depicted at the bottom left of \\Cref{fig:algorithm}.\n    This procedure can then be repeated\n    for the large operators supported on $A$ and $B$ to reduce the size of all the operators\n    involved, leading to the pattern in \\Cref{fig:algorithm} (a).\n    This reduces the problem of\n    implementing the time-evolution operator for $H$ into the problem of implementing smaller\n    time-evolution operators, which can be implemented using known quantum algorithms.\n    We now establish the lemmas needed to prove the result.\n    \n    \\begin{lemma}\n    Let $A_t$ and $B_t$ be continuous time-dependent Hermitian operators,\n    and let $U^A_t$ and $U^B_t$ with $U^A_0 = U^B_0 = {\\mathbf{1}}$ \n    be the corresponding time evolution unitaries.\n    Then the following hold:\n    \\begin{enumerate\n    \\item[(i)] $W_t=(U^B_t)^\\dagger U^A_t$ is the unique solution of \n    $i\\partial_t W_t = \\left((U^B_t)^\\dagger (A_t-B_t) U^B_t \\right) W_t$ \n    and $W_0 = {\\mathbf{1}}$.\n    \\item[(ii)] If $\\norm{A_s-B_s} \\le \\delta$ for all $s \\in [0,t]$,\n     then $\\norm{U^A_t - U^B_t} \\le t \\delta$.\n    \\end{enumerate}\n    \\label{lem:ode}\n    \\end{lemma}\n    \\begin{proof} \n    (i) Differentiate.\n    The solution to the ordinary differential equation is unique.\n    (ii) Apply Jensen's inequality for $\\norm{\\cdot}$ \n    (implied by the triangle inequality for $\\norm{\\cdot}$) \n    to the equation $W_t - W_0 = \\int_0^t {\\mathrm{d}} s \\partial_s W_s$.\n    Then, invoke (i) and the unitary invariance of $\\norm{\\cdot}$.\n    \\end{proof}\n\n    \\begin{lemma}[Lieb-Robinson \n    bound~\\cite{LiebRobinson1972,Hastings2004LSM,NachtergaeleSims2006,HastingsKoma2006}]\n    \\label{lem:LR}\\label{lem:LRB}\n    Let $H=\\sum_X h_X$ be a local Hamiltonian and \n    $O_X$ be any operator supported on $X$,\n    and put $\\ell = \\lfloor \\dist(X,\\Lambda\\setminus \\Omega) \\rfloor$.\n    Then\n    \\begin{align}\n    &\\norm{(U^{H}_t)^\\dagger O_X U^{H}_t\n        - (U^{H_{ \\Omega}}_t)^\\dagger O_X U^{H_{ \\Omega}}_t}\n    \\le \\nonumber \\\\\n    &|X|\\norm{O_X}\n    \\frac{(2\\zeta_0 |t|)^\\ell}{\\ell!},\n    \\label{eq:LRB}\n    \\end{align}\n    where $\\zeta_0 = \\max_{p \\in \\Lambda} \\sum_{Z \\ni p} |Z|\\norm{h_Z} = {{O}}(1)$.\n    In particular, there are constants $v_{LR} > 0$, called the Lieb-Robinson velocity,%\n    \\footnote{\n    Strictly speaking, the Lieb-Robinson velocity \n    is defined to be the infimum of any $v_{LR}$ such that \\cref{eq:exteriorLightcone} holds.\n    }\n    and $\\mu>0$, \n    such that for $\\ell \\geq v_{LR} |t|$, we have\n    \\begin{align}\n    &\\norm{(U^{H}_t)^\\dagger O_X U^{H}_t\n      - (U^{H_{ \\Omega}}_t)^\\dagger O_X U^{H_{ \\Omega}}_t} \\nonumber \\\\ \n      &\\le {{O}}(|X| \\norm{O_X} \\exp(- \\mu \\ell)).\n    \\label{eq:exteriorLightcone}\n    \\end{align}\n    \\end{lemma}\n    \\begin{proof}\n        See \\ref{app:pfLRB}.\n    \\end{proof}\n    We are considering strictly local interactions (as in \\Cref{thm:main}),\n    where $h_X = 0$ if $\\diam(X)>1$,\n    but similar results hold with milder locality conditions such as $\\norm{h_X} \\le e^{-\\diam(X)}$~\\cite{LiebRobinson1972,Hastings2004LSM,NachtergaeleSims2006,HastingsKoma2006,Hastings2010};\n    see the appendix for a detailed proof.\n    Below we will only use the result that \n    the error is at most ${{O}}(e^{ - \\mu \\ell})$ for some $\\mu > 0$ and fixed $t$.\n    For slower decaying interactions, the bound is weaker and\n    the overlap size $\\ell$ in \\Cref{fig:algorithm} will have to be larger.\n    \n    The Lieb-Robinson bound implies the following decomposition.\n    \\begin{lemma}\n    Let $H = \\sum_X h_X$ be a local Hamiltonian (as in \\Cref{thm:main}, or a more general definition\n    for which \\Cref{lem:LRB} still holds).\n    Then there are constants $v,\\mu >0$ such that\n    for any disjoint regions $A,B,C$, we have\n    \\begin{align}\n        &\\norm{U_t^{H_{A\\cup B}} (U_{t}^{H_{B}})^\\dagger U_t^{H_{B \\cup C}} \n        - U_t^{H_{A\\cup B \\cup C}}} \\le \\nonumber \\\\\n        &{{O}}( e^{vt - \\mu \\dist(A,C)} ) \\sum_{X: \\text{bd}(AB,C)}\\norm{h_X}\n    \\end{align}\n    where $X:\\text{bd}(AB,C)$ means that $X \\subseteq A \\cup B \\cup C$ \n    and $X \\not\\subseteq A \\cup B$  and $X \\not\\subseteq C$.\n    \\label{lem:decomposition}\n    \\end{lemma} \n    \\begin{proof}\n    We omit ``$\\cup$'' for the union of disjoint sets. \n    The following identity is trivial but important:\n    \\begin{align}\n        U^{H_{ABC}}_t =  \n        U^{H_{AB} + H_{C}}_t \\underbrace{(U^{H_{AB} + H_{C}}_t )^\\dagger  U^{H_{ABC}}_t}_{=W_t}.\n    \\end{align}\n    By \\Cref{lem:ode} (i), $W_t$ is generated by~\\cite{Osborne2006,Michalakis2012}\n    \\begin{align}\n    &(U^{H_{AB} + H_{C}}_t)^\\dagger \n    (\\underbrace{H_{ABC}-H_{AB}-H_C}_{H_\\text{bd}}) \n    U^{H_{AB} + H_{C}}_t \\nonumber \\\\\n    &\\quad = \n    (U^{H_{B} + H_{C}}_t)^\\dagger \n    H_\\text{bd} \n    U^{H_{B} + H_{C}}_t\n    +\n    \\underbrace{{{O}}(\\norm{H_\\text{bd}} e^{ v t -\\mu \\ell})}_{=\\delta} \\label{eq:truncation}\n    \\end{align}\n    where $\\ell$ \n    is the distance from the support of the boundary terms $H_\\text{bd}$ to $A$,\n    and the estimate of $\\delta$ is due to \\Cref{lem:LRB} \n    applied to local terms in $H_\\text{bd}$.\n    Since $H_\\text{bd}$ contains terms that cross between $AB$ and $C$,\n    the distance $\\ell$ is at least $\\dist(A,C)$ minus 2.\n\n    By \\Cref{lem:ode} (i) again,\n    the unitary generated by the first term of \\eqref{eq:truncation}\n    is $(U^{H_{B} + H_{C}}_t)^\\dagger U^{H_{BC}}_t$,\n    which can be thought of as the ``interaction picture'' time-evolution operator\n    of the Hamiltonian in \\eqref{eq:truncation}.\n    This is our simplification of the ``patching'' unitary,\n    which is $t \\delta$-close to $W_t$ by \\Cref{lem:ode} (ii).\n    \\end{proof}\n    \n    \\begin{proof}[Proof of \\protect{\\Cref{thm:main}}]\n    The circuit for simulating the Hamiltonian is described in \\Cref{fig:algorithm}.\n    The decomposition of time evolution unitary in \\Cref{fig:algorithm}\n    is a result of iterated application of \\Cref{lem:decomposition}.\n    For a one-dimensional chain, let $L$ be the length of the chain,\n    so there are ${{O}}(L)$ qubits.\n    Take a two contiguous blocks $Y$ and $Z$ of the chain that overlaps by length $\\ell \\ll L$.\n    Under periodic boundary conditions there are two components in the intersection,\n    and under  open boundary conditions,\n    there is one component in the intersection.\n    Applying \\Cref{lem:decomposition},\n    we decompose the full unitary into two blocks on $Y$ and $Z$, respectively,\n    and one or two blocks in the intersection.\n    Every block unitary in the decomposition is a time evolution operator\n    with respect to the sum of Hamitonian terms within the block,\n    and we can recursively apply the decomposition.\n    Making the final blocks as small as possible,\n    we end up with a layout of small unitaries as shown in \\Cref{fig:algorithm} (a).\n    The error from this decomposition is ${{O}}(\\delta L\/ \\ell)$,\n    which is exponentially small in $\\ell$ for $t = {{O}}(1)$.\n    \n    \n    Going to higher dimensions $D > 1$,\n    we first decompose the full time evolution \n    into unitaries on ${{O}}(L\/\\ell)$ hyperplanes (codimension 1).\n    This entails error ${{O}}(e^{-\\mu \\ell} L^D \/ \\ell)$ \n    since the boundary term has norm at most ${{O}}(L^{D-1})$.\n    For each hyperplane the decomposition into ${{O}}(L\/\\ell)$ blocks of codimension 2\n    gives error ${{O}}(e^{-\\mu \\ell} (\\ell L^{D-2}) (L\/\\ell))$.\n    Summing up all the hyperplanes, \n    we get ${{O}}(e^{-\\mu \\ell} L^{D}\/\\ell)$ for the second round of decomposition.\n    After $D$ rounds of the decomposition the total error is ${{O}}(e^{-\\mu \\ell} D L^{D}\/\\ell)$,\n    and we are left with ${{O}}((L\/\\ell)^D)$ blocks of unitaries for $t = {{O}}(1)$.\n    For longer times, apply the decomposition to each factor of \n    $U(T=t_M; t_{M-1})U(t_{M-1},t_{M-2}) \\cdots U(t_2;t_1)U(t_1; t_0 = 0)$\n    \n    It remains to implement the unitaries on $m = {{O}}(T L^D\/\\ell^D)$ blocks $\\square$ of ${{O}}(\\ell^D)$ qubits\n    where $\\ell = {{O}}(\\log (T L\/\\epsilon))$ to accuracy $\\epsilon \/ m$.\n    All blocks have the form $U^{H_{ \\square}}_t$, and can be implemented using any known\n    Hamiltonian simulation algorithm.\n    For a time-independent Hamiltonian, \n    if we use an algorithm that is polynomial in the spacetime volume\n    and polylogarithmic in the accuracy \n    such as those based on signal processing~\\cite{QSP,LC16} or \n    linear combination of unitaries~\\cite{BerryEtAl2014,TS,BCK15},\n    then the overall gate complexity is \n    ${{O}}(T L^D \\polylog(T L \/ \\epsilon)) = {{O}}(Tn \\polylog(Tn\/\\epsilon))$,\n    where the exponent in the $\\polylog$ factor depends on the choice of the algorithm.%\n    \\footnote{\n    If we use the quantum signal processing based algorithms~\\cite{QSP,LC16}\n    to implement the blocks of size ${{O}}(\\ell^D)$,\n    then we need ${{O}}(\\log \\ell)$ ancilla qubits for a block.\n    Thus, if we do not mind implementing them all in serial,\n    then it follows that the number of ancillas needed is \n    ${{O}}(\\log \\log (T n \/ \\epsilon))$,\n    which is much smaller than what would be needed if\n    the quantum signal processing algorithm was \n    directly used to simulate the full system.\n    }\n    For a slowly varying time-dependent Hamiltonian, we can use the \n    fractional queries algorithm~\\cite{BerryEtAl2014} or the Taylor series approach~\\cite{TS,Low2018,Kieferova2018} to\n    achieve the same gate complexity.\n    The Taylor series approach uses a subroutine\n    $\\ket t \\mapsto \\ket t \\left( \\sum_j \\abs{\\alpha_j(t)} \\right)^{-1\/2} \\sum_{j} \\sqrt{\\alpha_j(t)} \\ket j$,\n    where $\\alpha_j(t)$ is the real coefficient of Pauli operator $P_j$ in the Hamiltonian,\n    which must be efficiently evaluated.\n    \\end{proof}\n    For not too large system sizes $L$,\n    it may be reasonable to use a bruteforce method to decompose the block unitaries \n    into elementary gates~\\cite[Chap. 8]{KitaevBook}.\n    \n    \n    \n    \n\\section{Optimality}    \\label{sec:lowerbound}\n    \n    In this section we prove a lower bound on the gate complexity of problem of simulating the time evolution of a time-dependent local Hamiltonian. (Recall that throughout this paper we use \\emph{local} to mean geometrically local.) \n    \n\\subsection{Lower bound proofs}\n    \n    We now prove \\Cref{thm:lowerboundgeneral} and \\Cref{thm:lowerboundlocal}, starting with \\Cref{thm:lowerboundlocal}. This lower bound follows from the following three steps. First, in \\Cref{lem:circuittoHamiltonian}, we observe that for every depth-$T$ quantum circuit on $n$ qubits that uses  local $2$-qubit gates, there exists a Hamiltonian $H(t)$ of the above form such that time evolution due to $H(t)$ for time $T$ is equal to applying the quantum circuit. Then, in \\Cref{lem:countlower} we show that the number of distinct Boolean functions on $n$ bits computed by such quantum circuits is at least exponential in ${\\widetilde{\\Omega}}(Tn)$, where we say a quantum circuit has computed a Boolean function if its first output qubit is equal to the value of the Boolean function with high probability. Finally, in \\Cref{lem:countupper} we observe that the maximum number of Boolean functions that can be computed (to constant error) by the class of quantum circuits with $G$ arbitrary non-local $2$-qubit gates from any (possibly infinite) gate set is exponential in ${\\widetilde{\\calO}}(G\\log n)$. \n    Since we want this class of circuits to be able to simulate all piecewise constant bounded 1D Hamiltonians for time $T$, we must have $G = {\\widetilde{\\Omega}}(Tn)$.\n    \n    \\begin{lemma}\n        \\label{lem:circuittoHamiltonian}\n        Let $U$ be a depth-$T$ quantum circuit on $n$ qubits that uses local $2$-qubit gates from any gate set. \n        Then there exists a piecewise constant bounded 1D Hamiltonian $H(t)$ such that the time evolution due to $H(t)$ for time $T$ exactly equals $U$. \n    \\end{lemma}\n    \\begin{proof}\n        We first prove the claim for a depth-$1$ quantum circuit. This yields a Hamiltonian $H(t)$ that is defined for $t\\in[0,1]$, whose time evolution for unit time equals the given depth-$1$ circuit. Then we can apply the same argument to each layer of the circuit, obtaining Hamiltonians valid for times $t\\in[1,2]$, and so on, until $t\\in[T-1,T]$. This yields a Hamiltonian $H(t)$ defined for all time $t\\in[0,T]$ whose time evolution for time $T$ equals the given unitary. If the individual terms in a given time interval have bounded spectral norm, then so will the Hamiltonian defined for the full time duration. \n       \n        \n        For a depth $1$ circuit with local $2$-qubit gates, since the gates act on disjoint qubits we only need to solve the problem for one $2$-qubit unitary and sum the resulting Hamiltonians. Consider a unitary $U_j$ that acts on qubits $j$ and $j+1$. By choosing $H_j = i \\log U_j$, we can ensure that $e^{-iH_j} = U_j$ and $\\norm{H_j} = {{O}}(1)$. \n        \n        The overall Hamiltonian is now piecewise constant with $T$ time slices.\n    \\end{proof}\n    \n    Note that it also possible to use a similar construction to obtain a Hamiltonian that is continuous (instead of being piecewise constant) with a constant upper bound on the norm of the first derivative of the Hamiltonian.\n    \n    \\begin{lemma}\n        \\label{lem:countlower}\n        For any integers $n$ and $T$ such that $1 \\le n \\leq T \\leq 2^n$, the number of distinct Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ that can be computed by depth-$T$ quantum circuits on $n$ qubits that use local $2$-qubit gates from a finite gate set is at least $2^{{\\widetilde{\\Omega}}(Tn)}$.\n    \\end{lemma}\n    \\begin{proof}\n        We first divide the $n$ qubits into groups of $k=\\log_2 T$ qubits, which is possible since $T \\leq 2^n$. \n        On these $k$ qubits, we will show that it is possible to compute $2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions \n        with a depth $T$ circuit that uses local $2$-qubit gates. \n        One way to do this is to consider all Boolean functions on $k'<k$ bits. \n        Any Boolean function $f_x$ that evaluates to $f_x(x)=1$ on exactly one input $x$ of size $k'$ \n        can be computed with a circuit of ${\\widetilde{\\calO}}(k')$ gates and ${\\widetilde{\\calO}}(k')$ depth \n        using only $2$-qubit gates and $1$ ancilla qubit in addition to one output qubit~\\cite[Corollary 7.4]{BBC+95}.\n        An arbitrary Boolean function $f:\\{0,1\\}^{k'} \\to \\{ 0,1\\}$ is a sum of such functions: \n        $f = \\sum_{x \\in f^{-1}(1)} f_x = \\bigoplus_{x \\in f^{-1}(1)} f_x$.\n        Implementing all $f_x$ for $x \\in f^{-1}(1)$ in serial using a common output qubit,\n        we obtain a circuit for the full function $f$.\n        Since $f^{-1}(1)$ consists at most $2^{k'}$ bit strings, \n        this will yield a circuit of size ${\\widetilde{\\calO}}(2^{k'})$ and depth ${\\widetilde{\\calO}}(2^{k'})$.\n        Making all the gates local using SWAP gates does not change these expressions by more than a log factor in the exponent. \n        By choosing $k' = k - \\Theta(\\log k)$,\n        we can compute all Boolean functions on $k'$ bits with depth at most $T$. \n        Since there are $2^{2^{k'}} = 2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions on $k'$ bits, \n        we have shown that circuits with depth $T$ using $k=\\log_2 T$ qubits can compute at least $2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions.\n        \n        We can compute $2^{{\\widetilde{\\Omega}}(T)}$ distinct Boolean functions on each of the $n\/k$ blocks of $k$ qubits to obtain $(2^{{\\widetilde{\\Omega}}(T)})^{n\/k} = 2^{{\\widetilde{\\Omega}}(Tn)}$ distinct Boolean functions with $n\/k$ outputs. I.e., we have computed a function $\\{0,1\\}^n \\to \\{0,1\\}^{n\/k}$. Since we want to obtain a single-output Boolean function, as the overall goal is to prove lower bounds against simulation algorithms correct on local measurements, we combine these Boolean functions into one. We do this by computing the parity of all the outputs of these $n\/k$ Boolean functions using CNOT gates. Computing the parity uses at most $n$ 2-qubit local gates and has depth $n$. The circuit now has depth $T + n \\leq 2T$ and by rescaling $T$ we can make this circuit have depth $T$, while retaining the lower bound of $2^{{\\widetilde{\\Omega}}(Tn)}$ distinct Boolean functions.\n        \n        Unfortunately, after taking the parity of these $n\/k$ functions, it is not true that the resulting functions are all distinct. For example, the parity of functions $f(x)$ and $g(y)$ is a new function $f(x) \\oplus g(y)$, which also happens to be the parity of the functions $\\neg f(x)$ and $\\neg g(y)$. To avoid this overcounting of functions, we do not use all possible functions on $k'$ bits in the argument above, but only all those functions that map the all-zeros input to 0. This only halves the total number of functions, which does not change the asymptotic expressions above. With this additional constraint, it is easy to see that if $f(x) \\oplus g(y) = f'(x) \\oplus g'(y)$, this implies that $f$ and $f'$ are the same, by fixing $y$ to be the all-zeros input, and similarly that $g$ and $g'$ are the same.\n    \\end{proof}\n    \n    \n    We say a quantum circuit $U$ computes a Boolean function $f:\\{0,1\\}^n\\to\\{0,1\\}$ with high probability if measuring the first output qubit of $U|x_1x_2\\cdots x_n0\\cdots 0\\rangle$ yields $f(x)$ with probability at least $2\/3$.\n    \\begin{lemma}\n        \\label{lem:countupper}\n        The number of Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ that can be computed with high probability by quantum circuits with unlimited ancilla qubits using $G$ non-local $2$-qubit gates from any gate set is at most $2^{{\\widetilde{\\calO}}(G\\log n)}$.\n    \\end{lemma}\n    \\begin{proof}\n    First we note that if a circuit $U$ with $G$ arbitrary $2$-qubit gates from any gate set computes a Boolean function with probability at least $2\/3$, then there is another circuit with ${\\widetilde{\\calO}}(G)$ gates from a finite 2-qubit non-local gate set that computes the same function with probability at least $2\/3$. We do this by first boosting the success probability of the original circuit using an ancilla qubit to a constant larger than $2\/3$ and then invoking the Solovay--Kitaev theorem \\cite{NC00} to approximate each gate in this circuit to error ${{O}}(1\/G)$ with a circuit from a finite gate set of $2$-qubit gates. This increases the circuit size to ${\\widetilde{\\calO}}(G)$ gates. Since each gate has error ${{O}}(1\/G)$, the overall error is only a constant, and the new circuit approximates computes the Boolean function $f$ with high probability.\n        \n        We now have to show that the number of Boolean functions on $n$ bits computed by a circuit with ${\\widetilde{\\calO}}(G)$ non-local 2-qubit gates from a finite gate set is at most $2^{{\\widetilde{\\calO}}(G\\log n)}$. To do so, we simply show that the total number of distinct circuits with ${\\widetilde{\\calO}}(G)$ non-local 2-qubit gates from a finite gate set is at most $2^{{\\widetilde{\\calO}}(G \\log n)}$.\n        \n        First observe that a circuit with ${\\widetilde{\\calO}}(G)$ gates can only use ${\\widetilde{\\calO}}(G)$ ancilla qubits, since each $2$-qubit gate can interact with at most 2 new ancilla qubits. Furthermore, the depth of a circuit cannot be larger than the number of gates in the circuit. Let us now upper bound the total number of quantum circuits of this form. Each such circuit can be specified by listing the location of each gate and which gate it is from the finite gate set. Specifiying the latter only needs a constant number of bits since the gate set is finite, and the location can be specified using the gate's depth, and the labels of the two qubits it acts on. The depth only requires ${{O}}(\\log G)$ bits to specify, and since there are at most $n+{\\widetilde{\\calO}}(G)$ qubits, this only needs ${{O}}(\\log n + \\log G)$ bits to specify. In total, since there are ${\\widetilde{\\calO}}(G)$ gates, the entire circuit can be specified with ${\\widetilde{\\calO}}(G\\log n)$ bits. Finally, since any such circuit can be specified with ${\\widetilde{\\calO}}(G\\log n)$ bits, there can only be $2^{{\\widetilde{\\calO}}(G \\log n)}$ such circuits.\n    \\end{proof}\n    \n    \\begin{proof}[Proof of \\protect{\\Cref{thm:lowerboundlocal}}]\n        Suppose that any piecewise constant bounded 1D Hamiltonian on $n$ qubits \n        can be simulated for time $T$ using $G$ 2-qubit non-local gates from any (possibly infinite) gate set. \n        Then using \\Cref{lem:circuittoHamiltonian} and \\Cref{lem:countlower}, \n        we can compute at least $2^{{\\widetilde{\\Omega}}(Tn)}$ distinct Boolean functions using such Hamiltonians. \n        By assumption, each of these Boolean functions can be approximately computed by a circuit with $G$ gates. \n        Now invoking \\Cref{lem:countupper}, \n        we know that such circuits can compute at most $2^{{\\widetilde{\\calO}}(G\\log n)}$ $n$-bit Boolean functions. \n        Hence we must have $G\\log n = {\\widetilde{\\Omega}}(Tn)$, which yields $G = {\\widetilde{\\Omega}}(Tn)$.\n        \n        The proof for $T\\leq n$ follows in a black-box manner from the first part of the theorem statement by only using $T$ out of the $n$ available qubits. In this case the first part of the theorem applies and yields a lower bound of ${\\widetilde{\\Omega}}(T^2)$.\n    \\end{proof}\n    \n    We now prove \\Cref{thm:lowerboundgeneral}, which follows a similar outline. The first step is identical, and we can reuse \\Cref{lem:circuittoHamiltonian}. For the next step, instead of counting distinct Boolean functions, we count the total number of ``distinct'' unitaries. Unlike Boolean functions on $n$ bits, there are infinitely many  unitaries on $n$ qubits. Hence we count unitaries that are ``distinguishable.'' Formally, we say $U$ and $V$ are distinguishable if they are some constant, say $0.1$ for concreteness, apart in diamond norm. This is equivalent to the existence of a state $|\\psi\\rangle$ such that $U|\\psi\\rangle$ and $V|\\psi\\rangle$ have trace distance $0.1$. In \\Cref{lem:Ucountlower} we show that the number of distinguishable unitaries computed by quantum circuits on $n$ qubits with depth $T$ is exponential in ${\\widetilde{\\Omega}}(Tn)$. As before, we then show that the maximum number of distinguishable unitaries that can be computed (to constant error) by the class of quantum circuits with $G$ arbitrary non-local $2$-qubit gates from any (possibly infinite) gate set is exponential in ${\\widetilde{\\calO}}(G\\log n)$. \n    \n    \\begin{lemma}\n        \\label{lem:Ucountlower}\n        For any integer $n$ and integer $n\\leq T \\leq 4^n$, there exists a set of unitaries of size $2^{{\\widetilde{\\Omega}}(Tn)}$, such that every unitary in the set can be computed by a depth-$T$ quantum circuit on $n$ qubits that uses local $2$-qubit gates from a finite gate set, and the diamond norm distance between any $U \\neq V$ from this set is at least $0.1$. \n    \\end{lemma}\n    \\begin{proof}\n        We divide the $n$ qubits into groups of $k=\\log_4 T$ qubits, which is possible since $T \\leq 4^n$. On these $k$ qubits, we will \n        compute $2^{{\\widetilde{\\Omega}}(T)}$ distinguishable unitaries (i.e., unitaries that are at least distance $0.1$ apart from each other) with a depth $T$ circuit that uses local $2$-qubit gates. We can do this by considering a maximal set of unitaries on $k'$ qubits that is distinguishable. \n        More precisely, on $k'$ qubits there exist $2^{\\Omega(4^{k'})}$ unitaries \n        such that each pair of unitaries is at least distance $0.1$ apart; see e.g.~\\cite{Szarek1997}. \n        (This follows from the fact that in the group of $d \\times d$ unitaries with metric induced by operator norm, \n        a ball of radius $0.1$ has volume exponentially small in $d^2$.)\n        We know that any unitary on $k'$ qubits can be exactly written as a product of ${\\widetilde{\\calO}}(4^{k'})$ arbitrary 2-qubit gates~\\cite{BBC+95,Kni95}. Making these gates local and from a finite gate set only adds polynomial factors in $k'$. By choosing $k' = k - \\Theta(\\log k)$, we can compute this set of $2^\\Omega(4^{k'}) = 2^{{\\widetilde{\\Omega}}(T)}$ distinguishable unitaries with depth at most $T$. \n        \n        Since we can compute $2^{{\\widetilde{\\Omega}}(T)}$ pairwise-distant unitaries on each of the $n\/k$ blocks of $k$ qubits, we can compute $(2^{{\\widetilde{\\Omega}}(T)})^{n\/k} = 2^{{\\widetilde{\\Omega}}(Tn)}$ unitaries on all $n$ qubits by considering all possible combinations of unitaries on the different blocks. Finally, if $U$ and $V$ are distinguishable, then so are $U \\otimes X$ and $V \\otimes Y$, since the distinguisher can simply ignore the second register.\n    \\end{proof}\n    \n    \\begin{lemma}\n        \\label{lem:Ucountupper}\n        Let $S$ be a set of distinguishable unitaries (i.e., the diamond norm distance between any $U \\neq V$ from this set is at least $0.1$). Then if any unitary in $S$ can be computed by a quantum circuit with $G$ non-local $2$-qubit gates from any gate set, then $|S| = 2^{{\\widetilde{\\calO}}(G\\log n)}$.\n    \\end{lemma}\n    \\begin{proof}\n        This proof is essentially the same as that of \\Cref{lem:countupper}. We first observe that if a circuit over an arbitrary gate set computes a unitary $U$, we can approximate it to error less than $0.04$ using the Solovay--Kitaev theorem and increase the circuit size to ${\\widetilde{\\calO}}(G)$ gates. Importantly, since the unitaries are a distance $0.1$ apart, one circuit cannot simultaneously approximate two different unitaries to error $0.04$.\n        Then exactly the same counting argument as in \\Cref{lem:countupper} shows there can only be $2^{{\\widetilde{\\calO}}(G \\log n)}$ such circuits.\n    \\end{proof}\n    \n    \\begin{proof}[Proof of \\protect{\\Cref{thm:lowerboundgeneral}}]\n        Suppose that any piecewise constant bounded local Hamiltonian on $n$-qubits could be simulated for time $T$ using $G$ 2-qubit non-local gates from any (possibly infinite) gate set. Then using \\Cref{lem:circuittoHamiltonian} and \\Cref{lem:Ucountlower}, we can produce a set $S$ of distinguishable unitaries of size $2^{{\\widetilde{\\Omega}}(Tn)}$. By assumption, each of these unitaries can be approximately computed by a circuit with $G$ non-local 4-qubit gates. Now invoking \\Cref{lem:countupper}, we know that such circuits can approximate at most $2^{{\\widetilde{\\calO}}(G\\log n)}$ distinguishable unitaries on $n$ qubits. Hence we must have $G\\log n = {\\widetilde{\\Omega}}(Tn)$, which yields $G = {\\widetilde{\\Omega}}(Tn)$.\n    \\end{proof}\n    \n\\section{Discussion}\n    \n    We have only analyzed local Hamiltonians on (hyper)cubic lattices embedded in some Euclidean space,\n    but Lieb-Robinson bounds with exponential dependence on the separation distance\n    hold more generally.  However, on more general graphs, it may be more difficult to find an appropriate decomposition that gives a small error; this must be analyzed for each graph.\n    One advantage of the method here is that the accuracy improves for smaller Lieb-Robinson velocity.\n    This can occur if the terms in the Hamiltonian have a small commutator (see Appendix). \n    \n    The decomposition based on Lieb-Robinson bounds looks \n    very similar to higher order Lie-Trotter-Suzuki formulas.\n    The difference is in the fact that the overlap \n    is chosen to be larger and larger\n    (though very slowly) as the simulated spacetime volume increases.\n    If we want an algorithm that does not use any ancilla qubits,\n    similar to algorithms based on Lie-Trotter-Suzuki formulas,\n    then we can simulate the small blocks from Lieb-Robinson bounds\n    by high order Suzuki formulas~\\cite{Suzuki1991,BAC+07} \n    where the accuracy dependence is polynomial (power-law) of arbitrarily small exponent $a >0$.\n    This combination results in an algorithm of total gate complexity $O(Tn(Tn\/\\epsilon)^a)$,\n    similar to what is claimed to be achievable in Ref.~\\cite[Sec.~4.3]{JordanLeePreskill2014}.\n    \n    Application to fermions is straightforward but worth mentioning.\n    Since Hamiltonian terms always have fermion parity even,\n    Lieb-Robinson bounds hold without any change.\n    Given the block decomposition based on the Lieb-Robinson bound,\n    we should implement each small blocks in $\\polylog(Tn\/\\epsilon)$ gates.\n    The Jordan-Wigner transformation, a representation of Clifford algebra,\n    is a first method one may consider:\n    \\begin{align}\n    \\gamma_{2j-1} &\\mapsto \\sigma^z_1 \\otimes \\cdots \\otimes \\sigma^z_{j-1} \\otimes \\sigma^x_j,\\\\\n    \\gamma_{2j}   &\\mapsto \\sigma^z_1 \\otimes \\cdots \\otimes \\sigma^z_{j-1} \\otimes \\sigma^y_j,\n    \\end{align}\n    where $\\gamma_{2j-1},\\gamma_{2j}$ are Majorana (real) fermion operators,\n    and the right-hand side is a tensor product of Pauli matrices.\n    Often, the tensor factor of $\\sigma^z$ preceding $\\sigma^x$ or $\\sigma^y$ is called \n    a {\\it Jordan-Wigner string}.\n    In one spatial dimension, the above representation \n    where the ordering of $\\gamma$ is the same as the chain's direction\n    gives a local qubit Hamiltonian, since in any term Jordan-Wigner strings cancel.\n    The ordering of fermions is thus very important.\n    (Under periodic boundary conditions, at most one block may be nonlocal; \n    however, we can circumvent the problem \n    by regarding the periodic chain, a circle, \n    as a double line of finite length whose end points are glued:\n    $ \\big( [-1,+1] \\times \\{\\uparrow,\\downarrow\\} \\big) \/ \\{ (-1,\\uparrow) \\equiv (-1,\\downarrow), (+1,\\uparrow) \\equiv (+1,\\downarrow)\\} $.\n    This trick doubles the density of qubits in the system,\n    but is applicable in any dimensions for periodic boundary conditions.)\n    \n    In higher dimensions with fermions,\n    a naive ordering of fermion operators\n    turns most of the small blocks into nonlocal operators\n    under the Jordan-Wigner transformation.\n    However, fortunately, there is a way to get around this, at a modest cost,\n    by introducing auxiliary fermions and let them mediate the interaction of a target Hamiltonian~\\cite{Verstraete2005}.\n    The auxiliary fermions are ``frozen,'' during the entire simulation, \n    by an auxiliary Hamiltonian that commutes with the target Hamiltonian.\n    With a specific ordering for the fermions,\n    one can represent all the new interaction terms as local qubit operators.\n    The key is that if $c_j c_k$ is a fermion coupling whose Jordan-Wigner strings do not cancel,\n    we instead simulate $c_j c_k \\gamma_1 \\gamma_2$, where $\\gamma_{1,2}$ are auxiliary,\n    such that the Jordan-Wigner strings of $\\gamma_1,\\gamma_2$ cancel those of $c_j,c_k$, respectively.\n    The auxiliary $\\gamma$'s may be ``reused'' for other interaction terms \n    if the interaction term involves fermions that are close in a given ordering of fermions.\n    (Ref.~\\cite{Verstraete2005} explains the manipulation for quadratic terms\n    but it is straightforward that any higher order terms can be treated similarly.\n    They also manipulate the Hamiltonian for the auxiliary $\\gamma$ to make it local after Jordan-Wigner transformation,\n    but for our simulation purposes it suffices to initialize the corresponding auxiliary qubits.)\n    In this approach, if we insert ${{O}}(1)$ auxiliary fermion per fermion in the target system,\n    the gate and depth complexity is the same as if there were no fermions.\n    Note that we can make the density of auxiliary fermions arbitrarily small\n    by increasing the simulation complexity.\n    Divide the system with non-overlapping blocks of diameter $\\ell$, which is e.g. ${{O}}(\\log n)$,\n    that form a hypercubic lattice.\n    (These blocks have nothing to do with our decomposition by Lieb-Robinson bounds.)\n    Put ${{O}}(1)$ auxiliary fermions per block,\n    and order all the fermions lexicographically so that all the fermions in a block\n    be within a consecutive segment of length ${{O}}(\\ell^D)$ in the ordering.\n    Interaction terms within a block have Jordan-Wigner string of length at most ${{O}}(\\ell^D)$,\n    and so do the inter-block terms using the prescription of \\cite{Verstraete2005}.\n    The gate and depth complexity of this modified approach has $\\mathrm{poly}(\\ell)$ overhead.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nElectric tunnel junctions consist of two electrically conductive electrodes separated by an insulating tunnel barrier. In these tunnel junctions the barrier plays a crucial role for optimizing and engineering device properties. For example, by employing a barrier with a ferroelectric polarization it is possible to exploit the polarization state as a means to store information in the so called tunnel electroresistance states~\\cite{Garcia2014,Kim2013,Singh2015}, or utilizing a ferromagnetic barrier for spin filtering properties~\\cite{Moodera_2007}. In magnetic tunnel junctions (MTJs), consisting of two ferromagnetic electrodes, the tunnel magnetoresistance (TMR) can be further enhanced by utilizing wave function filtering of the tunnel barrier, as for example in Fe\/MgO\/Fe tunnel junctions with TMR ratios reaching up to $800\\;\\%$ at room temperature~\\cite{Zhang_Butler_2004,Parkin2004,Yuasa2004,Ikeda_TMR_2008}. However, for large TMR ratios not only the functional properties are relevant, but also a good lattice match between the electrode and barrier materials are necessary. This problem becomes apparent in MTJs based on Heusler electrodes with MgO barrier, where the lattice mismatch between electrodes and barrier reduces significantly the observed TMR~\\cite{ReviewHeusler_2012}.\n\nIn the following we investigate ferromagnetic La$_{0.67}$Sr$_{0.33}$MnO$_{3}$(LSMO) based MTJs utilizing a novel insulating barrier material: strontium tin oxide (SrSnO$_3$, SSO). SSO crystalizes in a cubic perovskite structure with a lattice constant $a=0.40254\\,\\mathrm{nm}$ (Ref.~\\onlinecite{SSO-crystal}). Up to now SSO has only been used as a transparent conductive layer, when doped with La, Ba,~\\cite{liu_structure_2012,kurre_studies_2011,wang_transparent_2010} as an insulation barrier with a high dielectric constant in single flux quantum circuits~\\cite{wakana_examination_2005}, and as a photoelectrochemical converter for the reduction of water~\\cite{bellal_visible_2009}. Our experiments show that it is possible to fabricate MTJs with an SSO barrier that exhibt large TMR ratios of up to $350\\:\\%$ at liquid He temperatures and large resistance-area products of up to $30\\;\\mathrm{M\\Omega \\mu m^2}$. We compare these results to reference MTJs with a SrTiO$_{3}$ (STO) barrier and compare the obtained results to already existing publications. We start with a short description of the experimental techniques used, then present the results of our experiments and conclude with a summary.\n\n\\section{Experiment}\n\nThe samples investigated here have been grown by pulsed laser deposition (PLD) from stoichiometric, ceramic LSMO, SSO and STO targets (99.99\\% purity) on (001)-oriented STO substrates. The pulsed laser deposition was carried out in an UHV-chamber with a base pressure of $2\\times10^{-7}\\;\\mathrm{Torr}$ using a KrF excimer laser ($248\\;\\mathrm{nm}$ wavelength, $2\\;\\mathrm{Hz}$ repetition rate) at a substrate temperature of $700^\\circ\\mathrm{C}$. For the LSMO, SSO, and STO deposition a laser fluence of $1.2\\;\\mathrm{J\/cm^2}$, $0.8\\;\\mathrm{J\/cm^2}$, and $1.2\\;\\mathrm{J\/cm^2}$ and an oxygen atmosphere $200\\;\\mathrm{mTorr}$, $200\\;\\mathrm{mTorr}$, and $100\\;\\mathrm{mTorr}$, respectively, has been used. After deposition the samples have been cooled down to room temperature in an oxygen atmosphere of $200\\;\\mathrm{mTorr}$\n\nTo determine the crystal phase and epitaxy of the resulting films, a standard 4 circle x-ray diffraction (XRD) setup (Phillips X'pert Pro) was used with a Cu $K\\alpha$ source. Transmission electron microscope (TEM) cross sectional samples were prepared by conventional mechanical polishing and Ar ion milling. The scanning TEM (STEM) imaging were carried out in an aberration-corrected Nion UltraSTEM 200 microscope operating at $200\\;\\mathrm{kV}$.\n\nFor the fabrication of the MTJs from the blanket films we utilized a three step photolithography process as detailed in Ref.~\\onlinecite{lu_large_1996}. In the first step, we defined the bottom contact mesa by an initial Ar ion milling etch followed by a chemical wet etch in diluted HCl (1:10) to avoid Ar-ion induced conductivity of the STO substrate.~\\cite{gross_situ_2011} After the etching process, a SiO$_2$ insulation layer was sputter deposited and the photoresist removed via lift-off. In the second step the tunnel junctions were formed by Ar ion milling and SiO$_2$ deposition. In the final step, top contacts were fabricated by Ru sputter deposition and lift-off.\n\nFor the electrical characterization the samples were mounted in the variable temperature insert of a Dynacool PPMS system at temperatures $5\\;\\mathrm{K} \\leq T \\leq 300\\;\\mathrm{K}$ and in magnetic fields $H$ applied in the film plane of up to $9\\;\\mathrm{T}$. A constant DC-bias voltage was applied to the MTJs, while recording the 4-pt DC-voltage drop $V_\\mathrm{4pt}$ using an Agilent HP 3478 voltmeter and the DC current flow $I$ using a Keithley 428 current amplifier and an Agilent HP 3458a voltmeter across the junction (See insert in Fig.~\\ref{figure:RT_LSMO_SSO}).\n\n\\section{Results and Discussion}\n\n\\begin{figure}[t,b]\n  \\includegraphics[width=85mm]{Fig1.pdf}\\\\\n  \\caption[XRD results for LSMO based Tunneljunctions]{(Color online) (a) Full range $2\\theta-\\omega$-scan for a LSMO(43)\/SSO(7.5)\/LSMO(17) trilayer on a STO (001) substrate. Only $(00l)$ reflections from STO, LSMO, and SSO are visible. (b) Enlarged scale of the $2\\theta-\\omega$-scan of (a) around the STO (002) reflection. (c) Rocking curve of the SSO(002) reflection exhibits a FWHM of $0.03^\\circ$ indicating a small mosaic spread.\n  \\label{figure:XRD_LSMO}}\n\\end{figure}\n\nWe first discuss the XRD data exemplarily obtained for a LSMO(43)\/SSO(7.5)\/LSMO(17) trilayer; here as throughout the text values in parenthesis are the corresponding film thicknesses in nm. The full range $2\\theta-\\omega$-scan in Fig.~\\ref{figure:XRD_LSMO}(a) exhibits no secondary phases. Only peaks that can be related to $(00l)$-reflections of the LSMO, SSO layers and the STO substrate are visible and even Laue oscillations around the SSO $(001)$-reflection are present. This indicates a good epitaxial (001)-oriented growth of SSO on LSMO. Moreover, in the high resolution $2\\theta-\\omega$-scan around the STO $(002)$-reflection the characteristic $K\\alpha_1$, $K\\alpha_2$ peak splitting of the LSMO peak indicates a high crystalline quality of the LSMO layers. A rocking curve of the LSMO $(002)$-reflection (not shown here) yields a full width at half maximum (FWHM) of the reflection peak of $0.02^\\circ$, which is close to the instrumental resolution of our setup. From the peak position of the SSO $(002)$-reflection we extract an out-of-plane lattice constant of $0.413\\;\\mathrm{nm}$, that is slightly larger then the bulk value of SSO, which we attribute to the compressive strain due to the growth on LSMO on a STO substrate (bulk lattice constant LSMO $a_\\mathrm{LSMO}=0.3894\\;\\mathrm{nm}$, bulk lattice constant STO $a_\\mathrm{STO}=0.3902\\;\\mathrm{nm}$, Ref.~\\onlinecite{Vailionis2011}). For the rocking curve of the SSO $(002)$-reflection shown in Fig.~\\ref{figure:XRD_LSMO}(c) we obtain a FWHM of $0.03^\\circ$, indicating a low mosaic spread of the SSO layer. We also employed $\\phi$-scans at the (101) and (202) reflections of substrate and deposited layers (data not shown here), which verified the in-plane epitaxial relation $[100]_\\mathrm{STO}\\parallel[100]_\\mathrm{SSO}\\parallel[100]_\\mathrm{LSMO}$. Taking all these findings together, our deposition parameters yield highly epitaxial LSMO\/SSO\/LSMO trilayers with excellent structural quality. Similar results have also been obtained with the very same growth conditions for our reference LSMO\/STO\/LSMO trilayers.\n\n\\begin{figure*}[b,t]\n  \\includegraphics[width=170mm]{Fig2.pdf}\\\\\n  \\caption[STEM images of SSO and STO trilayers]{(Color online)STEM Z-contrast images of $[110]_\\mathrm{pc}$-oriented (a) LSMO(43)\/SSO(7.5)\/LSMO(17) trilayer showing a dislocation in the SSO layer and (b) LSMO(50)\/STO(4)\/LSMO(30) trilayer. The colormaps below the Z-contrast images show the distribution of strain tensor components $\\epsilon_\\mathrm{xx}$ and $\\epsilon_\\mathrm{yy}$ in the respective images, where x and y are the in-plane and out-of-plane directions as shown by the arrows in (b). The scale bar in both the images is $2\\;\\mathrm{nm}$.\n  \\label{figure:TEM_LSMO}}\n\\end{figure*}\n\nWe now focus on the comparison of the structural quality of We now focus on the comparison of the structural quality of the two trilayers obtained from atomic resolution STEM. Figure~\\ref{figure:TEM_LSMO}(a) and (b) show high-angle annular dark field (HAADF) STEM images obtained for a LSMO(43)\/SSO(7.5)\/LSMO(17) and a LSMO(50)\/STO(4)\/LSMO(30) trilayer, respectively. Both the films are oriented along the $[110]_\\mathrm{pc}$ direction (pc denotes pseudocubic). As the contrast in a HAADF STEM image is roughly proportional to $Z^2$ (Ref.~\\onlinecite{Pennycook_2011}), with Z being the atomic number, the cation columns in both the trilayers can be clearly observed, while the lighter oxygen columns are invisible. Within the LSMO layers, the (La\/Sr)O layers appear brighter than the MnO$_2$ layers. Similarly in the STO barrier in Fig.~\\ref{figure:TEM_LSMO}(b), SrO layers appear brighter than TiO$_2$ layers. However, In the SSO barrier in Fig.~\\ref{figure:TEM_LSMO}(a), Sn (Z = 50) being heavier than Sr (Z = 38), the SnO$_2$ layers appear brighter than the SrO layers. While it is clear that the LSMO\/STO\/LSMO trilayer in Fig.~\\ref{figure:TEM_LSMO}(a) is free of any line defects and is of a high quality, the SSO layer and the LSMO\/SSO interfaces are defective with the presence of dislocations. We performed geometric phase analysis (GPA) of the two Z-contrast images to quantify the strain within the trilayers \\cite{Htch1998}, which are shown as colormaps below the respective images. For the LSMO\/SSO\/LSMO trilayer, GPA of $\\epsilon_\\mathrm{yy}$ component of the stress tensor shows presence of a very large dilation ($\\approx 9\\% $) in the out-of-plane direction in the SSO barrier, as expected due to the larger lattice constant of SSO. The $\\epsilon_\\mathrm{xx}$ component on the other hand shows large in-plane dilation and compression around the core of the dislocation, which is formed due to the lattice mismatch. In contrast, GPA of the STO barrier reveals only a small strain along the growth direction.\nFrom the TEM analysis we can conclude that the SSO barrier layer contains a higher density of defects, due to the larger lattice misfit between SSO and LSMO. This seems to be in contrast to our structural analysis via XRD, where we found excellent structural quality. This difference can be rationalized by the integral properties of XRD diffraction, while TEM reflects local properties of a sample. This highlights the importance of microstructural analysis via TEM imaging when analyzing the structural quality of samples.\n\n\\begin{figure}[b,t]\n  \\includegraphics[width=85mm]{Fig3.pdf}\\\\\n  \\caption[temperature dependence of resistance]{(Color online) Temperature dependence of the bottom contact resistance (black symbols) and of the MTJ resistance-area product (red line). The resistance of the bottom contact decreases with decreasing temperature. In contrast, the resistance-area product of the MTJ increases with decreasing $T$ (note the logarithmic scale), indicating that the resistance of the junction is dominated by the tunnel barrier and not the contact leads. The RA has been obtained under a bias voltage $V_\\mathrm{4pt}=100\\;\\mathrm{mV}$.\n  \\label{figure:RT_LSMO_SSO}}\n\\end{figure}\n\nIn the following we present our electrical transport data, focusing on the results obtained for a LSMO(43)\/SSO(2)\/LSMO(25) trilayer and a MTJ contact with a $2.5\\times2.5\\;\\mathrm{\\mu m^2}$ area. Note however, that similar results have been obtained for MTJs up to an area of $16\\;\\mathrm{\\mu m^2}$. A schematic drawing of the measurement setup is shown in the inset of Fig.~\\ref{figure:RT_LSMO_SSO}. We first discuss the temperature dependence of the resistance-area product (RA) of the MTJ obtained for a bias voltage $V_\\mathrm{4pt}=100\\;\\mathrm{mV}$ as depicted in Fig.~\\ref{figure:RT_LSMO_SSO}. Clearly, the RA increases with decreasing temperature $T$, while the LSMO bottom contact resistance measured simultaneously decreases with decreasing $T$. From this finding we conclude that the resistance of the MTJ is dominated by the tunnel barrier itself. In addition, the resistance of the junction is at least one order of magnitude larger than the bottom contact such that any magnetoresistance contributions from the bottom LSMO contact can be neglected in our measurements. Interestingly, the RA of MTJs based on a SSO barrier is rather large, reaching values of up to $30\\;\\mathrm{M\\Omega\\mu m^2}$ compared to values of up to $500\\;\\mathrm{k\\Omega\\mu m^2}$ obtained for the STO based reference junctions at $V_\\mathrm{4pt}=100\\;\\mathrm{mV}$. We attribute the increase to the larger bandgap of SSO ($4.1\\;\\mathrm{eV}$, Ref.~\\onlinecite{Zhang2006174}) as compared to STO ($3.3\\;\\mathrm{eV}$, Ref.~\\onlinecite{STO_bandgap_2001}), which directly leads to an increase in junction resistance. However, we would like to note that the high defect density in the SSO barrier might also play a crucial role for the RA, but would require further systematic studies.\n\n\\begin{figure}[b,t]\n  \\includegraphics[width=85mm]{Fig4.pdf}\\\\\n  \\caption[I-V characteristics at low T]{(Color online) (a) Bias dependence of the magnetoresistance at $T=5\\;\\mathrm{K}$ for a bias voltage of $0.2\\;\\mathrm{mV}$ (black), $22\\;\\mathrm{mV}$ (red), $42\\;\\mathrm{mV}$ (blue), and $100\\;\\mathrm{mV}$ (magenta) in a $2.5\\times2.5\\;\\mathrm{\\mu m^2}$ sized junction. The MR decreases with increasing bias voltage. Arrows in the figure indicate the sweep direction of the external magnetic field. (b) Current density vs applied bias voltage at $T=5\\;\\mathrm{K}$ for the very same sample as in (a) at two different external magnetic field values. For the electrode magnetization in the parallel state at $\\mu_0 H=-200\\mathrm{mT}$ (black symbols) a higher current flows as compared to the antiparallel arrangement at $\\mu_0 H=15\\mathrm{mT}$(red symbols). (c) Bias voltage dependence of the TMR extracted from the current-voltage characteristics of (b) (black line) and from the magnetic field sweeps at different $V_\\mathrm{4pt}$ of (a).\n  \\label{figure:IV_LSMO_SSO}}\n\\end{figure}\nAs a next step we look into the magnetic field-dependent magnetoresistance (MR) of the $2.5\\times2.5\\;\\mathrm{\\mu m^2}$ SSO junction at $T=5\\;\\mathrm{K}$. For this we measured $I(H)$ at fixed $V_\\mathrm{4pt}$ while sweeping the external applied magnetic field $H$ applied in the film plane along the bottom electrode strip from $\\mu_0 H=-200\\mathrm{mT}$ to $\\mu_0 H=200\\mathrm{mT}$ (upsweep) and back to $\\mu_0 H=-200\\mathrm{mT}$ (downsweep). From these measurements we calculated first the resistance $R(H)=V_\\mathrm{4pt}\/I(H)$ and then the magnetoresistance $MR(H)$ using\n\\begin{equation}\nMR(H)=\\frac{R(H)-R(-200\\;\\mathrm{mT})}{R(-200\\;\\mathrm{mT})}.\n\\label{equ:MR}\n\\end{equation}\nThe results of this procedure for different bias voltages $V_\\mathrm{4pt}$ are compiled in Fig.~\\ref{figure:IV_LSMO_SSO}(a). For all bias values used, we observe the typical pseudo spin-valve behaviour of a MTJ with two free magnetic layers: For large negative magnetic fields the magnetizations of the two LSMO electrodes are oriented parallel, which results in a low resistance state. When the external magnetic field is then increased to $\\mu_0 H=10\\;\\mathrm{mT}$ the bottom LSMO electrode changes its orientation, resulting in an (nearly) antiparallel alignment of the two magnetizations. We observe a high resistance state, yielding a positive MR. We verified that the first switching is indeed coming from the bottom electrode by an independent anisotropic magnetoresistance measurement of the bottom electrode. We find a nice agreement between the switching fields extracted from both independent measurements (data not shown here). By further increasing $H$ the magnetization of the top LSMO layer also changes its orientation resulting again in a parallel orientation of the two magnetizations and a low resistance state at $\\mu_0 H=670\\;\\mathrm{mT}$. A similar behaviour is observed when reversing the sweep direction only at negative fields. It is quite remarkable that bottom and top LSMO layer exhibit different switching fields. We attribute this to the difference in strain of the two layers induced by either the lattice mismatch between LSMO and SSO or the fabrication of the MTJs itself and the therefore resulting difference in lateral size for the top and bottom electrode. The difference in magnetic switching fields is further highlighted by the much steeper MR change occurring when the bottom electrode changes its orientation around $\\mu_0 H=10\\;\\mathrm{mT}$ and the more gradual change in MR when the top electrode starts to change into the parallel configuration for $30\\;\\mathrm{mT} \\leq \\mu_0 H \\leq 70\\;\\mathrm{mT}$. This may be explained by higher volume to surface ratio of the top LSMO electrode making it more susceptible to pinning defects, due to surface roughness, introduced by the MTJ fabrication process. In an independent series of experiments we measured the coercive fields of blanket LSMO films grown on (001)-oriented STO substrates with and without a $10\\;\\mathrm{nm}$ thick SSO layer between LSMO and substrate using a vibrating sample magnetometer (data not shown here). Within the experimental error we could not find any difference in the coercive fields for the different LSMO layers. These results suggest that the fabrication of the MTJs is an important factor for explaining the differences in switching fields for the top and bottom LSMO electrodes and further supports our assumption that pinning occurs due to the surface roughness introduced via the MTJ fabrication process.\n\nWe also studied the effect of the voltage bias on the observed $MR(H)$ for $V_\\mathrm{4pt}=0.2,22,42, 100\\;\\mathrm{mV}$ as shown in Fig.~\\ref{figure:IV_LSMO_SSO}(a) encoded in the graph as black, red, blue, and magenta symbols, respectively. The observed maximum MR increases with decreasing $V_\\mathrm{4pt}$. Such a behaviour should be expected if spin-flip scattering across the tunnel barrier and magnon scattering is enhanced for higher bias values.~\\cite{LeClair2005} In addition, the coercive field of the top electrode shifts to lower magnetic fields when increasing $V_\\mathrm{4pt}$. We attribute this to a slight temperature increase of the MTJ due to the higher current density flowing through it at larger voltage bias. Another possible explanation may be spin transfer torque due to the not perfectly antiparallel orientation of bottom and top electrode. However, to quantify this effect, further experiments have to be conducted, which go beyond the scope of this work.\n\nTo get a deeper insight into the bias dependence we also recorded I-V curves at a fixed applied magnetic field for parallel ($-200\\;\\mathrm{mT}$, black symbols) and antiparallel ($15\\;\\mathrm{mT}$) alignment of the ferromagnetic electrodes by changing $V_\\mathrm{4pt}$ and recording $I$ as shown in Fig.~\\ref{figure:IV_LSMO_SSO}(b). Comparing the two different I-V curves, it becomes apparent, that for antiparallel alignment the MTJ is in its high resistance state (lower current density for same bias voltage), while in the parallel state it is in its low resistance state (higher current density for same bias voltage). This finding is in agreement with the standard Julliere two spin current model and the difference in scattering rates due to different density of states at the fermi level for majority and minority spin carriers~\\cite{Julliere1975}. Moreover, we applied a Simmons fit~\\cite{Simmons1963} to the parallel magnetization state, which yields $3.2\\;\\mathrm{eV}$ barrier height and $2.6\\;\\mathrm{nm}$ barrier thickness for SSO. The barrier thickness agrees reasonably well with the $2\\;\\mathrm{nm}$ determined from X-ray reflectometry measurements. The determined barrier height is lower than the bulk bandgap of SSO ($4.1\\;\\mathrm{eV}$~\\cite{Zhang2006174}), which we attribute to the large concentration of defects in our SSO layer as observed in STEM images (see Fig.~\\ref{figure:TEM_LSMO}(a)). Moreover, the barrier height and bandgap of the material are not one and the same quantity: The barrier height will also depend on the position of the Fermi energy of the contact electrodes with respect to the bandgap, such that the barrier height should always be smaller than the band gap of the barrier material.\n\nGoing a step further we used the obtained I-V curves to directly calculate the bias dependence of the TMR (maximum of MR in Fig.~\\ref{figure:IV_LSMO_SSO}(a)) by determining the resistance for each bias voltage in the parallel ($R_\\mathrm{para}(V_\\mathrm{4pt})$ at $-200\\;\\mathrm{mT}$) and antiparallel ($R_\\mathrm{anti}(V_\\mathrm{4pt})$ at $15\\;\\mathrm{mT}$) alignment and using these values to calculate the TMR via\n\\begin{equation}\nTMR(V_\\mathrm{4pt})=\\frac{R_\\mathrm{anti}(V_\\mathrm{4pt})-R_\\mathrm{para}(V_\\mathrm{4pt})}{R_\\mathrm{para}(V_\\mathrm{4pt})}.\n\\label{equ:TMR}\n\\end{equation}\nThe obtained TMR bias dependence is shown in Fig.~\\ref{figure:IV_LSMO_SSO}(c) drawn as a black line. In addition, we included the MR obtained from the full $MR(H)$-loops [calculated also by Eq.~(\\ref{equ:TMR})] for the 4 different $V_\\mathrm{4pt}$ (Fig.~\\ref{figure:IV_LSMO_SSO}(a)) as black circles into Fig.~\\ref{figure:IV_LSMO_SSO}(c). We obtain a good agreement between the two differently determined TMR values for $V_\\mathrm{4pt}\\geq 42\\;\\mathrm{mV}$. For lower bias values both methods yield different results: While for the TMR determined from I-V curves we see a decrease of the TMR for low bias values, the TMR determined from $MR(H)$ loops monotonically increases with decreasing $V_\\mathrm{4pt}$. We attribute these discrepancies to small temperature fluctuations during the measurement of the I-V curves, which are most prominent at small bias values, leading to the observed bias dependence. Thus, we assume that we in reality observe an increase of the TMR with decreasing $V_\\mathrm{4pt}$. As already mentioned above, this behaviour is observed, when spin-flip and magnon scattering play a dominant role and influences on the TMR due to the band structure of the two ferromagnets can be neglected. A similar bias dependence has been observed in our STO based reference MTJs and also in other publications with STO as a tunnel barrier~\\cite{lu_large_1996}. The largest TMR value of $350\\;\\%$ in our junction for $T=5\\;\\mathrm{K}$ is obtained at $V_\\mathrm{4pt}= 0.2\\;\\mathrm{mV}$. Similar values have already been obtained in LSMO based MTJs with STO as a tunnel barrier, for example Garcia et al.~\\cite{garcia_temperature_2004} reported a maximum TMR of $540\\;\\%$ at $4.2\\;\\mathrm{K}$, while Sun et al.~\\cite{Sun1996} reported a maximum TMR of $\\approx200\\;\\%$ at $4.2\\;\\mathrm{K}$. For LSMO junctions with STO barriers the TMR could be enhanced by up to $1900\\;\\%$ at low temperatures by carefully optimizing growth, layer structures and patterning process of the MTJs~\\cite{Bowen2003,Werner2011}. This suggests that the TMR obtained in LSMO based MTJs with a SSO barrier could be further enhanced by further optimization. However, the TMR in these junctions is already comparable to values obtained with a STO barrier, despite the large density of defects present in the SSO barrier as determined by TEM analysis.\n\n\\begin{figure}[t,b]\n  \\includegraphics[width=85mm]{Fig5.pdf}\\\\\n  \\caption[TMR as a function of T]{(Color online) Temperature dependence of the TMR at fixed bias voltage $V_\\mathrm{4pt}= 100\\;\\mathrm{mV}$. (a) $MR(H)$ loops obtained at $T=5,10,25,50,75,100,150,200\\;\\mathrm{K}$, depicted with different coloured circles (see legend in the graph). (b) Temperature dependence of the TMR extracted from $MR(H)$. The TMR decreases with increasing $T$, for $T>200\\;\\mathrm{K}$ the TMR completely vanishes.\n  \\label{figure:TMRTemp_LSMO_SSO}}\n\\end{figure}\n\nWe investigated the temperature dependence of the TMR by measuring $I(H)$ loops at fixed $V_\\mathrm{4pt}= 100\\;\\mathrm{mV}$ and $T$. From these measurements we determined the $R(H)$ and $MR(H)$ via Eq.\\eqref{equ:MR}, the results of this procedure are depicted in Fig.~\\ref{figure:TMRTemp_LSMO_SSO}(a). The MR decreases with increasing temperature, for temperatures larger than $200\\;\\mathrm{K}$ no MR was visible within the noise limit of our setup. The vanishing of the MR at high temperatures can be attributed to a possible dead magnetic LSMO layer at the interface between LSMO and SSO, similar as already discussed in literature for LSMO\/STO interfaces.~\\cite{Sun1999,Bibes2001} Moreover, the switching field of the top LSMO electrode decreases from $\\mu_0H=60\\;\\mathrm{mT}$ at $T=5\\;\\mathrm{K}$ to $\\mu_0H=24\\;\\mathrm{mT}$ at $T=200\\;\\mathrm{K}$. In contrast, the switching field of the bottom electrode changes only slightly from $\\mu_0H=10\\;\\mathrm{mT}$ at $T=5\\;\\mathrm{K}$ to $\\mu_0H=6\\;\\mathrm{mT}$ at $T=200\\;\\mathrm{K}$. We attribute this different evolution of the switching fields with temperature, as already discussed above, to the difference in strain and contributions of magnetic pinning defects for the top and bottom LSMO layer. The contributions of both effects will be reduced when increasing the temperature, leading to a reduction of the switching field of the top LSMO electrode.\n\nFrom the $MR(H)$-loops at different $T$ we extracted the TMR as a function of temperature by taking the maximum value of each $MR(H)$ curve at each different $T$. The resulting TMR temperature dependence is depicted in Fig.~\\ref{figure:TMRTemp_LSMO_SSO}(b). Overall, the TMR decreases with increasing $T$ and vanishes for $T>200\\;\\mathrm{K}$. However, the TMR extracted for $T=10\\;\\mathrm{K}$ clearly deviates from the general monotonic decreasing trend and is significantly lower than for $T=25\\;\\mathrm{K}$. We attribute this difference to the fact that the $T=10\\;\\mathrm{K}$ data has been obtained in a second cooling cycle, which might have changed the magnetic domain configuration at the interface.~\\cite{Werner2011}\n\n\\section{Conclusions}\nIn summary, we have investigated for the first time the measurement of TMR in MTJs consisting of LSMO electrodes and SSO as the barrier. Our results suggest that SSO could be a promising barrier material for MTJs, with quite good insulating behaviour even when a high density of defects is present in the barrier. Our results show that the observed TMR in LSMO based MTJs with SSO as a barrier is comparable to the results obtained with STO as a barrier. However, further improvements in the SSO barrier properties, especially reducing the number of defects by lowering the lattice mismatch between the magnetic electrodes and the SSO barrier, are necessary to fully explore the potential of SSO as a barrier for MTJs. A very promising approach might be the usage of better lattice matched magnetic electrode materials, for example certain full Heusler compounds. For Co$_2$FeAl a very small lattice mismatch is obtained as $\\sqrt{2}a_\\mathrm{SSO}=0.5693\\;\\mathrm{nm}$ is extremely close to the bulk lattice constant of Co$_2$FeAl ($a_\\mathrm{Co_2FeAl}=0.569\\;\\mathrm{nm}$ Ref.~\\cite{Ebke2010}). We thus expect SSO to represent a very interesting material choice for further MTJ experiments.\n\\begin{acknowledgments}\nWe gratefully acknowledge financial support via NSF-ECCS Grant No.~1102263. Work at ORNL was supported by the U.S.~Department of Energy (DOE) Office of Science, Office of Basic Energy Sciences, Materials Science and Engineering Directorate.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}