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+{"text":"\\section{Introduction}\n\nAn interacting electron system could be in the strongly interacting collision-dominated hydrodynamic regime where frequent inter-particle collisions lead to local thermodynamic equilibrium so that the system should be thought of as manifesting a collective hydrodynamic behavior rather than the usual collisionless behavior of weak interactions where the standard independent quasiparticle picture applies. The analogy is to real liquids (e.g. water) whose macroscopic long wavelength flow properties hardly depend on the microscopic details of the molecular interactions which only serve to determine the macroscopic hydrodynamic parameters such as viscosity. Although the possibility of electron hydrodynamics was suggested a long time ago~\\cite{Gurzhi:1968hydrodynamic}, there has been a great deal of recent interest in the subject arising from the possibility of the experimental observation of electron hydrodynamics in solid state materials~\\cite{Lucas:2018hydrodynamics}. The current work focuses on a specific theoretical question regarding electron hydrodynamics. \n\n\nThe question we address is the interplay of the electronic plasmon mode with the hydrodynamic sound mode: How does the plasmon affect the hydrodynamic sound mode and does the electron system in the hydrodynamic regime undergo collective plasmon oscillations at all? \n\nThe theoretical question was recently addressed for 2D metals in a recent work~\\cite{Lucas:2018electronic}, and we generalize the theory to 3D metals. The 3D generalization is nontrivial and is of interest because of the true long-range nature of electronic Coulomb interaction in 3D metals, leading to a collective plasmon mode with a finite energy even in the long wavelength limit~\\cite{Bohm:1953collective}. This gapped massive nature of 3D plasmons arises from the 3D Coulomb coupling going as $1\/q^2$, where $q$ is the wavenumber (or momentum). By contrast, 2D Coulomb coupling goes as $1\/q$, leading to the 2D plasmon going at long wavelength as $q^{1\/2}$. Since the hydrodynamic sound mode (the so-called 'first sound' in the helium literature), which is the electron analog of the ordinary acoustic sound, goes linear in $q$ at long wavelength by definition, the interplay of a gapped 3D plasmon going as $\\mathcal{O}(q^0)$ with hydrodynamic sound going as $\\mathcal{O}(q)$ is more intriguing than the interplay between the hydrodynamic sound and the 2D plasmon, both of which vanish at long wavelength albeit with different momentum scaling. If the hydrodynamic sound mode in 3D metals develops a long wavelength gap by virtue of Coulomb interaction, this becomes reminiscent of the Higgs mechanism with a linearly dispersing Goldstone mode acquiring a mass by virtue of long-range coupling although the electron hydrodynamics problem does not involve any symmetry breaking (or an underlying Higgs field) in order to produce the gapped sound mode. In fact, this is precisely what happens in a metallic superconductor where the expected linearly dispersing Goldstone mode associated with the spontaneous breaking of the gauge symmetry acquires a long wavelength gap becoming effectively the plasmon mode in the presence of long-range Coulomb coupling instead of the usual zero sound acoustic mode as in a neutral superfluid where there is no long-range Coulomb coupling~\\cite{Anderson:1958random, Anderson:1963plasmons, Prange:1963dielectric, Fertig:1990collective, Fertig:1991collective}.\n\nWe show in this work that indeed 3D Coulomb coupling leads to a mass or a gap in the hydrodynamic sound mode in 3D metals, and the sound mode becomes the effective long wavelength plasmon mode in the hydrodynamic regime of 3D metals.  This is, however, true only in the leading order in momentum, and the next-to-leading-order dispersion corrections in wavenumber are different for 3D plasmons in the collisionless regime and the first sound in the hydrodynamic regime. We also study the damping or decay of the plasmon mode (which is akin to the zero sound mode in the helium literature) in the collisionless regime and the hydrodynamic first sound mode in the collision-dominated regime. Our terminology in the paper uses 'collisionless' to imply the non-hydrodynamic regime of weak inter-particle collisions (where the standard 'plasmon' or zero sound mode exists). By contrast, the collision-dominated regime is the hydrodynamic regime of rapid inter-particle collisions where the first sound mode exists. We study both regimes including effects of 3D long-range Coulomb interaction.\n\nThe hydrodynamic description applies when the momentum conserving inelastic electron-electron interaction is much stronger than any other momentum relaxing elastic scattering mechanisms which might be present in the system. In metals, such momentum non-conserving scattering processes arise from electron-impurity and electron-phonon scattering, which typically dominate at low and high temperatures respectively, making the hydrodynamic regime difficult to realize experimentally in the laboratory although, in principle, a very clean metal should manifest electron hydrodynamics at very low temperatures where the phonon scattering rate (in terms of resistivity) is suppressed as $T^5$ and the quasiparticle scattering rate goes as $T^2$, where $T$ is the temperature~\\cite{Ashcroft:1976solid}. Eventually, at low enough temperatures hydrodynamics is cut off in metals by any residual impurity scattering. In a metal with negligible impurity and phonon scattering, electron-electron interactions should produce hydrodynamic behavior at long wavelength and low frequency. \n\n\\section{Summary of the main results} \\label{sec:summary}\n\nWe extend the theory of~\\cite{Lucas:2018electronic} to three dimensions. Namely, we construct a solvable model that admits the exact calculation of sound modes in both the collisionless regime and hydrodynamic regime. The sound mode in the collisionless regime is the zero sound mode or the plasmon, and the sound mode in the hydrodynamic regime is the usual sound mode (i.e., the first sound). In this section, we represent the main results from the solvable model, including the effect of long-range Coulomb interaction on the sound modes. We leave the detailed derivation of the results to the following sections, but the exactly solvable model already demonstrates the physics clearly.\n\nIn this exactly solvable model consisting of spinless fermions, we assume spherical symmetry and consider the nonvanishing Landau parameter only in the s-wave channel. We denote the dimensionless Landau parameter $F_0$ and we consider repulsive interactions so that $F_0>0$. The sound mode is highly damped when the interaction is attractive, and may even lead to instabilities~\\cite{Nozieres:2018theory}. In any case, our interest is 3D metals in which the direct electron-electron interaction is repulsive. These considerations make the calculations analytically tractable, while maintaining the essential physics. \n\nWe first present the result in the clean limit where the sound mode is not damped by impurity scattering, $\\gamma_\\text{imp} = 0$.  $\\gamma_\\text{imp}$ is the scattering rate between quasiparticles and impurities. The zero sound in the collisionless regime $\\omega \\gg \\gamma$, where $\\gamma$ denotes the scattering rate between quasiparticles, is given by\n\\begin{eqnarray} \\label{eq:zeroFull}\n&& \\omega = \\pm v_0 q - \\mathfrak{i} \\gamma \\frac{(F_0+1)^2 - 2(F_0-1) (\\frac{v_0}{v_F})^2 - 3 (\\frac{v_0}{v_F})^4 }{F_0 [ F_0 +1 - (\\frac{v_0}{v_F})^2]}, \\\\  && \\frac{v_0}{v_F} \\text{arccoth} \\frac{v_0}{v_F} = 1 + \\frac1{F_0}.\n\\end{eqnarray}\nwhere the second equation determines the zero sound velocity. This equation is valid for all $F_0>0$. For the weakly interacting Fermi liquid, $F_0 \\ll 1$, and the zero sound velocity is nonperturbative in interaction strength, namely, $v_0 = v_F (1 + e^{-2\/F_0})$. It approaches the Fermi velocity as one should anticipate for the free electron gas. For $F_0 > 1$, the dispersion can be simplified as\n\\begin{eqnarray} \\label{eq:zero}\n\t\t\\omega = \\pm \\sqrt{\\frac{F_0}3 + \\frac35} v_F q - \\mathfrak{i} \\gamma \\frac{2(5F_0 + 21)}{5F_0(5F_0 + 3)},\n\\end{eqnarray}\nwhere $v_F$ is the Fermi velocity. Decreasing the frequency, the system enters the collision-dominated hydrodynamic regime where $ \\omega \\ll \\gamma$. Then the zero sound crosses over smoothly to the first sound,\n\\begin{eqnarray} \\label{eq:first}\n\\omega = \\pm \\sqrt{ \\frac{F_0}3 + \\frac13} v_F q - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2.\n\\end{eqnarray}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6cm]{velocity} \n\t\\caption{\\label{fig:velocity}The zero sound and first sound velocities as a function of the Landau parameter, where the Fermi velocity is set to 1 as the unit. The zero sound velocity is larger than the first sound velocity for all $F_0>0$.}\n\\end{figure}\nThe linear term in momentum reveals the sound velocity. In Fig.~\\ref{fig:velocity}, one can see that the collisionless zero sound velocity is always larger than the hydrodynamic first sound velocity, $v_0>v_1$. For the asymptotically large Landau parameter, the following inequality holds true, too, i.e.,\n\\begin{eqnarray}\n\\frac{v_0}{v_1} = \\sqrt{\\frac{F_0 + 9\/5}{F_0+1}} > 1 . \n\\end{eqnarray}\nThe imaginary part leads to the damping of sound modes. In an interacting Fermi liquid, the scattering rate at low temperatures is given by $\\gamma \\propto T^2$. A crucial difference between the zero sound and the first sound is that the damping rate is proportional to the interaction scattering rate, $\\text{Im}(\\omega) \\propto\\gamma$, for zero sound while it is inversely proportional to the interaction scattering rate, $\\text{Im}(\\omega) \\propto \\gamma^{-1}$, for the first sound. These results are well known and also experimentally observed in normal He-3~\\cite{Abel:1966propagation}. Reproducing these results presents a consistency check of our solvable model.\n\nWhen the impurity scattering is strong $\\gamma_\\text{imp} > v_F q$, both sound modes will be damped by impurity scattering at small momentum. The propagating wave transitions to a quadratic diffusion mode\n\\begin{eqnarray}\n\t\\omega = - \\mathfrak{i} \\frac{v_1^2}{\\gamma_\\text{imp}}  q^2, \\quad v_1 = v_F \\sqrt{\\frac{1+F_0}3}\n\\end{eqnarray}\nand a fully gapped one $ \\omega = -\\mathfrak{i} \\gamma_\\text{imp}$. When the impurity scattering is weak $\\gamma_\\text{imp} < v_F q$, its effect is an additional damping of the zero sound mode (in addition to the damping induced by quasiparticle interactions $\\gamma$); that is, the correction is $ \\delta\\text{Im}(\\omega) = \\frac12 \\gamma_\\text{imp}$, also see~(\\ref{eq:weak_imp}).\n\nNow we discuss the effect of Coulomb interaction on the sound mode. It is well known that the plasmon is fully gapped in 3D metals because of Coulomb interaction~\\cite{Bohm:1953collective}. Since both the plasmon and the sound wave are density fluctuating collective modes in many-body systems, it is naturally expected that the sound mode should also develop a finite gap due to the Coulomb interaction. Since Coulomb interaction acts at the s-wave channel, one can make the following replacement,\n\\begin{eqnarray}\n\tF_0 \\rightarrow F_0 + \\frac{8\\pi \\alpha}{\\lambda_F^2 q^2},\n\\end{eqnarray}\nwhere $\\alpha = \\frac{e^2}{4\\pi v_F}$ is the effective finite structure constant for the 3D metal ($v_F$ is the Fermi velocity), and $\\lambda_F = \\frac{2\\pi}{k_F}$ is the Fermi wavelength.\nThe Coulomb interaction [the second term in (6)] should be understood as the effective one felt by quasiparticles, where the effective fine structure constant captures all possible renormalization effects in going from bare electrons to quasiparticles. This is a generalization from the neutral Fermi liquid theory to the Landau-Silin theory of a charged Fermi liquid~\\cite{Zala2001:interaction}.\nSince we are interested in the long wavelength limit, where the Coulomb interaction dominates over any short-range interaction, we focus on~(\\ref{eq:zero}) in the appropriate limit. After making this replacement, the zero sound becomes\n\\begin{eqnarray} \\label{eq:zeroPlas}\n\t\\omega = \\pm \\left( \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F} + \\frac{9\/5+F_0}{4\\sqrt{6\\pi \\alpha}} v_F \\lambda_F q^2 \\right) - \\mathfrak{i} \\gamma \\frac{\\lambda_F^2 q^2}{20 \\pi \\alpha},\n\\end{eqnarray}\nand the first sound becomes\n\\begin{eqnarray} \\label{eq:firstPlas}\n\t\\omega = \\pm \\left( \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F} + \\frac{1+F_0}{4\\sqrt{6\\pi \\alpha}} v_F \\lambda_F q^2 \\right) - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2.\n\\end{eqnarray}\n\nIt is instructive to compare the results with the plasmon mode. The 3D plasmon dispersion is well known and is determined by the following equation in RPA calculations,\n\\begin{eqnarray} \\label{eq:plasmon3D}\n\t\\omega^2 = \\omega_0^2 + \\frac35 \\frac{k_F^2}{m^2} \\frac{\\omega_0^2}{\\omega^2} q^2, \\quad \\omega_0^2 = \\frac{e^2 N_e}{m},\n\\end{eqnarray}\nwhere $N_e = \\frac{4\\pi}3 \\frac{k_F^3}{(2\\pi)^3} = \\frac{k_F^3}{6\\pi^2}$ is the electron density in 3D,  $m = \\frac{k_F}{v_F}$ is the effective mass of the quasiparticle, and $\\omega_0$ is the well-known plasmon frequency. (Note a conventional difference between the plasmon frequency in our~(\\ref{eq:plasmon3D}) and Eq.~(14) in Ref.~[4\n, which comes from the Coulomb potential we define as~(\\ref{eq:energy}) having an extra factor of $1\/4\\pi$ in real space as we use the rationalized unit instead of the Gauss unit.) If we expand the plasmon mode at long wavelength, its dispersion reads\n\\begin{eqnarray} \\label{eq:3Dplasmon}\n\t\\omega = \\omega_0 + \\frac35 \\frac{k_F^2}{m^2 \\omega_0} \\frac{q^2}2 =  \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F} + \\frac{9\/5}{4\\sqrt{6\\pi \\alpha}} v_F \\lambda_F q^2 , \\nonumber \\\\\n\\end{eqnarray}\nwhere in the second step, we change the parameters to better compare with sound mode results. Comparing~(\\ref{eq:3Dplasmon}) to the zero sound mode with Coulomb interaction~(\\ref{eq:zeroPlas}), we find a correction to the quadratic dispersion from the Landau parameter $F_0$ and a quadratic damping due to the collisions. Comparing~(\\ref{eq:3Dplasmon}) to the first sound mode with Coulomb interaction~(\\ref{eq:firstPlas}), we conclude that in the hydrodynamic regime, the plasmon dispersion gets corrected by a factor of $5\/9$ at the next-to-leading $q^2$ order because of Coulomb coupling. Again there is a damping effect in~(\\ref{eq:firstPlas}) inherited from the first sound mode.\n\nMore generally, we can consider a formal long-range interaction given by a power law defined by $q^{-2\\eta}$, where $\\eta=1$ for 3D Coulomb coupling, i.e.,\n\\begin{eqnarray}\n\tF_0 \\rightarrow F_0 + \\frac{8\\pi \\alpha}{(\\lambda_F q)^{2\\eta}},\n\\end{eqnarray}\nwhere $\\alpha$ is the effective interaction strength, and the parameter $\\eta$ defines the form of the interaction. In real space, this translates to a potential of the form $r^{2\\eta - 3}$. For $\\eta >1$ ($\\eta < 1$) it is stronger (weaker) than the Coulomb potential. This type of interaction leads to the zero sound mode\n\\begin{eqnarray}\n\\omega = \\pm \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F^\\eta} q^{1-\\eta} - \\mathfrak{i} \\gamma \\frac{(\\lambda_F q)^{2\\eta}}{20 \\pi \\alpha},\n\\end{eqnarray}\nand the first sound mode\n\\begin{eqnarray}\n\\omega = \\pm \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F^\\eta} q^{1-\\eta} - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2.\n\\end{eqnarray}\nIn this case, the sound mode has an interesting dispersion resulting from the long-range interaction. Also note that the damping rate for the \"hydrodynamic plasmon\" does not change due to the specific form of the long-range interaction, remaining independent of the range parameter $\\eta$.\n\n\\section{Review of the Boltzmann equation in a 3D metal}\n\nThe collective mode is one of the fundamental excitations of a many-body system. It emerges from coherent interactions between quasiparticles, and is fundamentally different from single particle electron-hole type excitations. It is convenient to describe the collective mode by the distribution function $n(\\textbf k, \\textbf r, t)$ of the quasiparticle at given momentum $\\textbf k$ and position $\\textbf r$. As we are interested in the effect of Coulomb interaction on the sound mode, we will restrict ourselves to the spinless electron. Spin can be added straightforwardly at the cost of making the notations cumbersome---we emphasize that the modes we are discussing are charge density collective excitations which are independent of electron spin. The Boltzmann equation governing the dynamics of the distribution function is~\\cite{Nozieres:2018theory}\n\\begin{eqnarray}\n\t\\frac{d n(\\textbf k, \\textbf r, t)}{dt} = \\left( - \\frac{d \\textbf r}{dt} \\cdot \\frac{\\partial}{\\partial \\textbf r} - \\frac{d \\textbf k}{dt} \\cdot \\frac{\\partial}{\\partial \\textbf k} \\right) n(\\textbf k, \\textbf r, t) + \\mathcal{I}[n], \\nonumber \\\\\n\\end{eqnarray}\nwhere the first term on the right hand side is the drift term while the second term is the collision integral. The semiclassical equation of motion of a quasiparticle is well known\n\\begin{eqnarray}\n\t\\frac{d \\textbf k}{d t} &=& - \\frac{\\partial \\epsilon(\\textbf k, \\textbf r, t)}{\\partial \\textbf r}, \\\\\n\t\\frac{d \\textbf r}{d t} &=&  \\frac{\\partial \\epsilon(\\textbf k, \\textbf r, t)}{\\partial \\textbf k},\n\\end{eqnarray}\nwhere $\\epsilon(\\textbf k, \\textbf r, t)$ is the energy of the quasiparticle. Since we are interested in a conventional metal, it is sufficient to consider the semiclassical description without external electromagnetic field or Berry curvature. The quasiparticle energy should be determined self-consistently from the Boltzmann equation. We are going to solve the collective mode of a small variation from the Fermi-Dirac distribution function, namely,\n\\begin{eqnarray}\n\tn(\\textbf k, \\textbf r, t) = n_F(\\textbf k) + \\delta n(\\textbf k, \\textbf r, t), \\\\\n\tn_F(\\textbf k) \\equiv n_F[\\epsilon_0(\\textbf k)] = \\frac1{e^{\\beta(\\epsilon_0(\\textbf k) -\\mu)}+1},\n\\end{eqnarray}\nwhere $n_F(\\textbf k)$ is the Fermi-Dirac distribution at equilibrium, $\\epsilon_0(\\textbf k)$ is the bare energy of free electrons, $\\beta \\equiv 1\/T$ is the inverse temperature, and $\\mu$ denotes the chemical potential.\n\nThe total energy of the system with a small variation from equilibrium is\n\\begin{eqnarray} \n\t\\epsilon_{\\text{tot}}(t) &=& \\int d^3 \\textbf r \\int_{\\textbf k} \\epsilon_0(\\textbf k) \\delta n(\\textbf k, \\textbf r, t) \\nonumber\\\\\n\t&& + \\frac12 \\int d^3 \\textbf r \\int_{\\textbf k, \\textbf k'} \\delta n(\\textbf k, \\textbf r, t) f(\\textbf k, \\textbf k') \\delta n(\\textbf k', \\textbf r, t) \\nonumber\\\\\n\\label{eq:energy}\t&& + \\frac12 \\int d^3 \\textbf r d^3 \\textbf r' \\int_{\\textbf k, \\textbf k'} \\delta n(\\textbf k, \\textbf r, t) \\frac{e^2}{4\\pi}\\frac{1}{|\\textbf r - \\textbf r'|} \\delta n(\\textbf k', \\textbf r', t), \\nonumber\\\\\n\\end{eqnarray}\nwhere $\\int_{\\textbf k} \\equiv \\int \\frac{d^3 \\textbf k}{(2\\pi)^3}$ and $f(\\textbf k, \\textbf k')$ is the Landau parameter characterizing the short-range quasiparticle interactions, and the second line represents the long-range Coulomb interaction. \nThe presence of both short-range and long-range interactions is an important generalization to a charged Fermi liquid from the neutral Fermi liquid theory. \nThis generalization is also often referred to as Landau-Silin Fermi liquid theory.\nNote that throughout the paper, we use the rationalized unit, where the factor of $\\frac1{4\\pi}$ appears in the real space Coulomb potential.\nThus, by varying with respect to $\\delta n(\\textbf k, \\textbf r, t)$, we can get the quasiparticle energy\n\\begin{eqnarray}\n\t\\epsilon(\\textbf k, \\textbf r, t) =  \\epsilon_0(\\textbf k) + \\int_{\\textbf k'} f(\\textbf k, \\textbf k') \\delta n(\\textbf k', \\textbf r, t) \\nonumber \\\\\n\t+ \\int_{\\textbf k'} \\int d^3 \\textbf r' \\frac{e^2}{4\\pi} \\frac{1}{|\\textbf r - \\textbf r'|} \\delta n(\\textbf k', \\textbf r', t).\n\\end{eqnarray}\n\nTo get a wave-like collective mode, we assume in the usual manner that the variation takes the plane wave form\n\\begin{eqnarray}\n\t\\delta n(\\textbf k, \\textbf r, t) = \\delta n(\\textbf k) e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t}.\n\\end{eqnarray}\nActually, once we get the eigenmode from the plane wave expansion, we can construct any arbitrary mode using linear superposition and completeness. The energy for such a plane wave excitation is\n\\begin{eqnarray}\n\t\\epsilon(\\textbf k, \\textbf r, t) &=& \\epsilon_0(\\textbf k) + \\int_{\\textbf k'} f(\\textbf k, \\textbf k') \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t} \\nonumber\\\\ \n\t&& + \\int_{\\textbf k'} \\int d^3 \\textbf r' \\frac{e^2}{4\\pi}\\frac{1}{|\\textbf r - \\textbf r'|} \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r' - \\mathfrak{i} \\omega t} \\\\\n\t&=& \\epsilon_0(\\textbf k) + \\int_{\\textbf k'} \\left( f(\\textbf k, \\textbf k') +\\frac{e^2}{\\textbf q^2} \\right) \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t} , \\nonumber\\\\\n\\end{eqnarray}\nwhere in the second line, we have used the Fourier transform of the Coulomb potential in 3D. A simple derivation is given in Appendix~\\ref{append:Coulomb}. \n\nNow with the quasiparticle energy, the semiclassical equation of motion becomes\n\\begin{eqnarray}\n\t\\frac{d \\textbf k}{dt} &=& - \\mathfrak{i} \\textbf q \\int_{\\textbf k'} \\left( f(\\textbf k, \\textbf k') + \\frac{e^2}{\\textbf q^2}  \\right) \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t}, \\\\  \\frac{d \\textbf r}{dt} &=& \\frac{\\partial \\epsilon_0(\\textbf k)}{\\partial \\textbf k} \\equiv \\textbf v(\\textbf k).\n\\end{eqnarray}\nPutting this equation of motion into the Boltzmann equation, we arrive at\n\\begin{eqnarray}\n\t\\omega \\delta n(\\textbf k) = \\textbf q \\cdot \\textbf v(\\textbf k) \\Big( \\delta n(\\textbf k) - \\big[\\partial_\\epsilon\\big|_{\\epsilon = \\epsilon_0(\\textbf k)} n_F(\\epsilon) \\big] \\nonumber \\\\\n\t\\times \\int_{\\textbf k'} \\big(f(\\textbf k, \\textbf k') + \\frac{e^2}{\\textbf q^2} \\big) \\delta n(\\textbf k') \\Big) + \\mathfrak{i} \\mathcal{I}[n],\n\\end{eqnarray}\nwhere $\\partial_\\epsilon\\big|_{\\epsilon = \\epsilon_0(\\textbf k)}$ means taking the derivative with respect to $\\epsilon$ and then setting $\\epsilon= \\epsilon_0(\\textbf k)$. Since at low temperatures, the small variation $\\delta n(\\textbf k)$ concentrates near the Fermi surface according to the factor \n\\begin{eqnarray}\n\t\\lim_{\\beta \\rightarrow \\infty} -\\partial_\\epsilon\\big|_{\\epsilon = \\epsilon_0(\\textbf k)} n_F(\\epsilon) =\\delta\\big(\\epsilon_0(\\textbf k) - \\mu\\big),\n\\end{eqnarray}\nwhich tends to a delta function localized at the Fermi surface, it is easy to see that the solution to the equation has the form $\\delta n(\\textbf k) = - [\\partial_\\epsilon|_{\\epsilon = \\epsilon_0(\\textbf k)} n_F(\\epsilon)] \\delta n(\\sigma) = \\delta(\\epsilon_0(\\textbf k) - \\mu) \\delta n(\\sigma)$. Here, $\\sigma = (\\theta, \\phi) $ is determined by the vector $\\textbf k$ at the Fermi surface. Hence, it is convenient to change the variable from $\\textbf k$ to $\\big( \\epsilon_0(\\textbf k), \\sigma \\big) $. With the help of the identity,\n\\begin{eqnarray}\n\t\\int d^3 \\textbf k = \\int_0^\\infty d \\epsilon \\int_{\\epsilon_0(\\textbf k) = \\epsilon} \\frac{d\\sigma}{|\\textbf v(\\textbf k)|}, \n\\end{eqnarray}\nwhere $d \\sigma$ denotes the measure over the Fermi surface, and $d \\epsilon$ denotes the measure perpendicular to the Fermi surface, we arrive at\n\\begin{eqnarray} \\label{eq:Boltzmann}\n\t&& \\omega \\delta n(\\sigma) = \\textbf q \\cdot \\textbf v_F(\\sigma) \\Big( \\delta n(\\sigma) \\nonumber \\\\ \n\t&& + \\frac1{(2\\pi)^3} \\int \\frac{d\\sigma'}{|\\textbf v_F(\\sigma')|} \\big(f(\\sigma, \\sigma')  + \\frac{e^2}{\\textbf q^2} \\big) \\delta n(\\sigma') \\Big) + \\mathfrak{i} \\mathcal{I}[n],\n\\end{eqnarray}\nwhere due to the delta function, the integration is restricted to the Fermi surface, and we use $\\textbf v_F(\\sigma)$ to denote $\\textbf v(\\textbf k)$ when $\\textbf k$ is located at the Fermi surface which is the Fermi velocity, and we also use $f(\\sigma, \\sigma')$ to denote $f(\\textbf k, \\textbf k')$ when both $\\textbf k$ and $\\textbf k'$ are on the Fermi surface. The Coulomb interaction is independent of the angle variable and $\\textbf q$ is not a dynamical quantity, so one can regard the Coulomb interaction as a modification of the Landau parameter in the s-wave channel.\n\nThe Boltzmann equation~(\\ref{eq:Boltzmann}) is the central equation that governs the collective modes in a Fermi liquid, including sound modes. It can describe the situation with either short-range interaction or long-range interaction, treating the zero sound, first sound, and plasmon equivalently within one formalism. In the next section, we construct a simple model where the Boltzmann equation~(\\ref{eq:Boltzmann}) can be solved exactly.\n\n\n\\section{A solvable model}\n\nTo proceed, let us assume the Fermi liquid in question has spherical symmetry. This is the situation in simple 3D metals. As a result the Fermi velocity is independent of angle but $|\\textbf v_F(\\sigma)|= v_F$ and we can choose $\\textbf q = (0,0, q)$ pointing along the $k_z$ direction, and use the spherical coordinate $\\Omega=(\\theta, \\phi)$. Then (\\ref{eq:Boltzmann}) becomes\n\\begin{eqnarray} \\label{eq:Boltzmann_sph}\n\t&& \\omega \\delta n(\\Omega) = q v_F \\cos\\theta \\Big( \\delta n(\\Omega) \\nonumber\\\\\n\t&& + \\int \\frac{d\\Omega'}{4\\pi}\\left( F(\\Omega, \\Omega') + \\frac{8\\pi \\alpha}{\\lambda_F^2 \\textbf q^2} \\right) \\delta n(\\Omega') \\Big) + \\mathfrak{i} \\mathcal{I}[n],\n\\end{eqnarray}\nwhere we have used $\\int d \\sigma = k_F^2 \\int d\\Omega = k_F^2 \\int \\sin \\theta d\\theta d \\phi$, $k_F $ is the Fermi momentum $\\epsilon_0(k_F) =\\mu $ and $\\lambda_F$ is the corresponding Fermi wavelength $\\lambda_F = \\frac{2\\pi}{k_F}$. $\\alpha = \\frac{e^2}{4\\pi v_F}$ is the effective fine structure constant in the Fermi liquid defining the interaction coupling strength. \n$F(\\Omega, \\Omega') \\equiv \\frac{k_F^2 }{2\\pi^2 v_F} f(\\Omega, \\Omega')$ is the dimensionless Landau parameter arising from changing the variables from momenta to angles at the Fermi surface.\nAs we mentioned in the previous section, the Coulomb interaction is not a dynamical quantity in the Boltzmann equation. We can absorb the Coulomb interaction into the Landau parameter in the s-wave channel, and restore it back at the end of the calculation. \n\nIn the following, we assume that the only nonvanishing component of the Landau parameter is in the s-wave channel, namely, the Landau parameter is a constant $F(\\Omega, \\Omega') = F_0 $. We absorb the Coulomb interaction into the Landau parameter. This should be sufficient for our purpose to investigate the effect of the Coulomb interaction on the sound mode in a solvable mode. Without the collision integral, the Boltzmann equation reduces to the following eigen equation,\n\\begin{eqnarray} \\label{eq:Boltzmann_zero} \n\t(x_0 - \\cos \\theta) \\delta n(\\Omega) = F_0 \\int \\frac{d\\Omega'}{4\\pi}  \\delta n(\\Omega') , \\quad x_0 = \\frac{\\omega}{q v_F},\n\\end{eqnarray}\nwhich can solved~\\cite{Nozieres:2018theory} by \n\\begin{eqnarray} \\label{eq:n0}\n\\delta n(\\Omega) = \\frac{\\cos \\theta}{x_0 - \\cos \\theta}, \\quad x_0 \\text{arccoth} x_0  = 1+  \\frac1{F_0}.\n\\end{eqnarray}\nSince we ignore the collision integral, this solution represents, by definition, the zero sound solution in the collisionless regime. The second equation determines the velocity of the zero sound. When the $F_0 >1$, we get the approximate zero sound velocity given by $v_0 = \\sqrt{\\frac{F_0}3 + \\frac35} $. \n\n\nSince (\\ref{eq:Boltzmann_zero}) has spherical symmetry, we can study the eigen equation using the spherical harmonics. Any solution $\\delta n(\\Omega)$ can be expanded in the basis of spherical harmonics,\n\\begin{eqnarray}\n\t\\delta n (\\Omega) = \\sum_{l=0}^\\infty \\sum_{m=-l}^l \\delta n_{lm} Y_l^m(\\theta, \\phi),\n\\end{eqnarray}\nwhere $Y_l^m(\\theta, \\phi)$ is the spherical harmonic and $\\delta n_{lm}$ is the corresponding expansion coefficient. The sound modes are solutions with zero magnetic quantum number $m=0$. As the eigen equation conserves the magnetic quantum number, we consider the $m=0$ sector where the equation can be cast into\n\\begin{eqnarray}\n\tx_0 \\delta n_{l,0} &=& \\frac{l}{\\sqrt{4l^2-1}} \\delta n_{l-1,0} + \\frac{l+1}{\\sqrt{4(l+1)^2-1}} \\delta n_{l+1,0} \\nonumber\\\\ && + \\frac{F_0}{\\sqrt3} \\delta n_{0,0} \\delta_{l,1}.\n\\end{eqnarray}\nThe detailed derivation of this equation is given in Appendix~\\ref{append:math}. For $l \\ge 2$, it resembles a recurrence relation of a series. Indeed, it is not hard to check that for $l\\ge 2$, the following series of hypergeometric functions satisfies the recurrence relation,\n\\begin{eqnarray} \\label{eq:series}\n\ta_{l,m} (x) &=& 2\\pi \\delta_{m,0} \\sqrt{\\frac{2l+1}{4}}  \\frac{1}{(2x)^l} \\frac{\\Gamma(l+1)}{\\Gamma(l+ \\frac32)} \\nonumber\\\\\n\t&&\\times \\ _2F_1\\left( \\frac{l+1}2 , \\frac{l+2}2; l + \\frac32; \\frac1{x^2}\\right), ~ l \\ge 1.\n\\end{eqnarray}\nThis series is consistent with the eigenfunction (\\ref{eq:n0}). We present the details of obtaining the series in Appendix~\\ref{append:math}. Thus, we have $\\delta n_{l,m} = a_{l,m}(x_0)$ for $l \\ge 1$. The two equations for $l=1$ and $l=0$ read \n\\begin{eqnarray}\n\tx_0 \\delta n_{1,0} &=& \\frac{1+F_0}{\\sqrt 3} \\delta n_{0,0} + \\frac2{\\sqrt{15}} \\delta n_{2,0},  \\\\\n\tx_0 \\delta n_{0,0} &=& \\frac{1}{\\sqrt{3}} \\delta n_{1,0},\n\\end{eqnarray}\nwhich lead to the following solution,\n\\begin{eqnarray}\n\t\\delta n_{0,0} &=&  2\\sqrt \\pi (x_0 \\text{arccoth} x_0 -1), \\\\\n\t\\quad x_0 \\text{arccoth} x_0  &=& 1+  \\frac1{F_0}.\n\\end{eqnarray}\nThis is of course consistent with the previous eigen-solution~(\\ref{eq:n0}).\n\nTo access the hydrodynamic regime, where the collisionless zero sound crosses over to the hydrodynamic first sound, the collision integral plays an essential role. We take the collision integral to have the following form~\\cite{Lucas:2018electronic}\n\\begin{eqnarray}\n\t\\mathcal{I}(n) = -   \\sum_{l=0}^\\infty \\gamma_l \\delta n_{l,0}, \\quad \t\\gamma_l = \\begin{cases}\n\t\t\t\t0 & l = 0\\\\\n\t\t\t\t\\gamma_\\text{imp} & l = 1 \\\\\n\t\t\t\t\\gamma & l \\ge 2\n\t\t\t\t\\end{cases}.\n\\end{eqnarray}\nwhere $\\gamma$ is the collision rate from collisions between quasiparticles and $\\gamma_\\text{imp}$ is the collision rate between quasiparticles and impurities. Obviously, $\\gamma$ is the key hydrodynamic interaction parameter. Because collisions between quasiparticles conserve the particle number and their total momentum, the pure quasiparticle collision rate for $l=0, 1$ vanishes by virtue of conservation laws. On the other hand, elastic collisions between quasiparticles and quenched impurities relax the momentum, and, therefore, $\\gamma_\\text{imp}$ is nonzero for $l=1$. With this collision integral, the Boltzmann equation in the basis of spherical harmonics reduces to the following coupled equations,\n\\begin{eqnarray}\n\tx \\delta n_{l,0} &=& \\frac{l}{\\sqrt{4l^2-1}} \\delta n_{l-1,0} + \\frac{l+1}{\\sqrt{4(l+1)^2-1}} \\delta n_{l+1,0} , \\nonumber\\\\\n\t && \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad l \\ge 2, \\\\\n\tx_\\text{imp} \\delta n_{1,0} &=& \\frac{1 + F_0}{\\sqrt 3} \\delta n_{0,0} + \\frac2{\\sqrt{15}} \\delta n_{2,0},  \\\\\n\tx_0 \\delta n_{0,0} &=& \\frac{1}{\\sqrt{3}} \\delta n_{1,0}, \\\\\n\\label{eq:x}\tx_0 &=& \\frac{\\omega}{q v_F},~ x = \\frac{\\omega + \\mathfrak{i} \\gamma}{q v_F}, ~ x_\\text{imp} = \\frac{\\omega + \\mathfrak{i} \\gamma_\\text{imp}}{q v_F}.\n\\end{eqnarray}\nThe series~(\\ref{eq:series}) again solves the first recurrence relation, with $x$ replacing $x_0$. Thus for $l \\ge 1$, $\\delta n_{l,m} = a_{l,m}(x) $. The other two equations can be solved easily, namely, $\\delta n_{0,0} = 2 \\sqrt \\pi \\frac{x}{x_0} (x \\text{arccoth} x - 1)$ and\n\\begin{eqnarray} \\label{eq:eigen}\n (F_0 + 1 -3 x_0 x_\\text{imp}) (x \\text{arccoth} x - 1) \\nonumber\\\\ + x_0 [(3x^2-1) \\text{arccoth} x - 3x ] = 0.\n\\end{eqnarray}\nThis is the main result of our paper. The eigen dispersion of the sound mode is determined by~(\\ref{eq:eigen}), where the definitions of $x$'s are given by~(\\ref{eq:x}).\n\n\nTo investigate the sound mode, we can assume $\\gamma_\\text{imp} = 0$, otherwise the coherent propagating sound mode is damped. In the collisionless regime, $\\omega \\gg \\gamma$, we obtain the zero sound mode or the plasmon in~(\\ref{eq:zeroFull}) as\n\\begin{eqnarray}\n\t&& \\omega = \\pm v_0 q - \\mathfrak{i} \\gamma \\frac{(F_0+1)^2 - 2(F_0-1) (\\frac{v_0}{v_F})^2 - 3 (\\frac{v_0}{v_F})^4 }{F_0 [ F_0 +1 - (\\frac{v_0}{v_F})^2]}, \\\\ && \\frac{v_0}{v_F} \\text{arccoth} \\frac{v_0}{v_F} = 1 + \\frac1{F_0}.\n\\end{eqnarray}\nIn the case of strong repulsion $F_0 > 1$ (this is also the case for Coulomb interaction in the long wavelength limit), the sound mode can be simplified to\n\\begin{eqnarray}\n\t\\omega = \\pm \\sqrt{\\frac{F_0}3 + \\frac35} v_F q - \\mathfrak{i} \\gamma \\frac{2(5F_0 + 21)}{5F_0(5F_0 + 3)},\n\\end{eqnarray}\nwhich is nothing but~(\\ref{eq:zero}). On the other hand, in the collision-dominated hydrodynamic regime $\\omega \\ll \\gamma$ we get the first sound,\n\\begin{eqnarray}\n\t\\omega = \\pm \\sqrt{ \\frac{F_0}3 + \\frac13} v_F q - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2,\n\\end{eqnarray}\nwhich is~(\\ref{eq:first}).\n\nWe can also consider the effect of the finite impurity scattering. For weak impurity scattering, the effect is to modify the damping defined by imaginary parts of the sound mode. For the zero sound, we have\n\\begin{eqnarray} \\label{eq:weak_imp}\n\\omega &=& \\pm v_0 q - \\mathfrak{i} \\gamma \\frac{(F_0+1)^2 - 2(F_0-1) (\\frac{v_0}{v_F})^2 - 3 (\\frac{v_0}{v_F})^4 }{F_0 [ F_0 +1 - (\\frac{v_0}{v_F})^2]} \\nonumber\\\\\n&& - \\mathfrak{i} \\gamma_\\text{imp} \\frac{3(\\frac{v_0}{v_F})^2[(\\frac{v_0}{v_F})^2-1]}{F_0(F_0 +1 -(\\frac{v_0}{v_F})^2)}.\n\\end{eqnarray}\nFor large $F_0 > 1$, this reduces to a simple correction $\\delta \\text{Im}(\\omega) = \\frac12 \\gamma_\\text{imp}$. \nFor strong impurity scattering $\\gamma_\\text{imp} > q$, both sound modes are over-damped into\n\\begin{eqnarray}\n\t\\omega = - \\mathfrak{i} \\frac{F_0+1}{3\\gamma_\\text{imp}} v_F^2 q^2, \\qquad \\omega = -\\mathfrak{i} \\gamma_\\text{imp}.\n\\end{eqnarray}\n\t\n \n\n\\section{Conclusions} \\label{sec:conclusion}\n\nWe have discussed electronic sound modes in 3D metals in the presence of long-range Coulomb coupling via the Boltzmann equation. A more microscopic approach to the collective mode, like plasmons, would be to start from the electron Hamiltonian with long-range interactions. Then the collective mode results from integrating out the electron fluctuations. In the lowest order, this is nothing but the RPA approach. Here, we recapitulate how it works for 3D metals. In the RPA approximation, the collective mode is determined by the following eigen equation,\n\\begin{eqnarray} \\label{eq:RPA}\n\t1- V(\\textbf q) \\Pi(\\textbf q, \\omega) = 0,\n\\end{eqnarray}\nwhere $V(\\textbf q)$ is the interaction at momentum $\\bf q$, and the dynamical electron polarization function is defined by\n\\begin{eqnarray}\n\t&& \\Pi(\\textbf q, \\omega) = \\frac{1}{\\beta} \\sum_n \\int_{\\textbf k} \\frac1{-\\mathfrak{i} \\Omega_n - \\frac\\omega2+ \\xi(\\textbf k + \\frac{\\textbf q}2)} \\frac1{-\\mathfrak{i} \\Omega_n + \\frac\\omega2+ \\xi(\\textbf k - \\frac{\\textbf q}2)}, \\nonumber \\\\\n\t\\\\\n\t&& \\Omega_n = \\frac{(2n+1)\\pi}{\\beta}, \\quad \\xi(\\textbf k) = \\epsilon_0(\\textbf k) - \\mu.\n\\end{eqnarray}\nwhere $\\Omega_n$ is the Matsubara frequency and $\\omega$ is the real frequency. To proceed, we assume a spherical Fermi surface with parabolic dispersion $\\epsilon_0(\\textbf k) = \\frac{\\textbf k^2}{2m}$. It is not hard to get the 3D polarization function at zero temperature, namely,\n\\begin{eqnarray} \\label{eq:polarization3D}\n\t\\Pi(\\textbf q, \\omega) = \\frac{N_e q^2}{m \\omega^2} \\left( 1 + \\frac35 \\frac{k_F^2}{m^2} \\frac{q^2}{\\omega^2} \\right), \\quad N_e = \\frac{4\\pi}3 \\frac{k_F^3}{(2\\pi)^3},\n\\end{eqnarray}\nwhere $q \\equiv |\\textbf q|$. Putting in the Coulomb potential given by $V_\\text{Cou}(q) = \\frac{e^2}{q^2}$ and the polarization function~(\\ref{eq:polarization3D}) into eigen equation~(\\ref{eq:RPA}), we obtain the conventional plasmon given in~(\\ref{eq:plasmon3D}). \n\nWe can also consider a short-range density-density interaction potential that is independent of momentum, namely,\n\\begin{eqnarray} \n\tV({\\bf q}) = \\frac{2\\pi^2 v_F}{ k_F^2} F_0,\n\\end{eqnarray}\nwhere the prefactor originates from how the Landau parameter is introduced via quasiparticle interactions at the Fermi surface, i.e., $\\int_{F.S.} \\frac{d^3 {\\bf q}}{(2\\pi)^3}  V({\\bf q})...= \\int \\frac{d\\Omega}{4\\pi} (\\frac{ k_F^2}{2\\pi^2 v_F} V(\\Omega))... $. For the s-wave channel, it reduces to the above equation.\n\nUsing this interaction potential to replace the Coulomb potential, we get the following linearly dispersing sound-like collisionless collective mode \n\\begin{eqnarray}\n\t\\omega^2 = \\frac{v_F F_0}{4\\pi k_F^2} \\frac{N_e q^2}{m} \\left( 1 + \\frac35 \\frac{k_F^2}{m^2} \\frac{q^2}{\\omega^2} \\right), \\quad \\omega \\approx \\pm \\sqrt{\\frac{F_0}3 + \\frac35} v_F q. \\nonumber \\\\\n\\end{eqnarray}\nThe second equation is exactly the zero sound mode in~(\\ref{eq:zero}). At zero temperature, within the RPA, there is no damping of the electronic collective mode by quasiparticle collisions since the collision rate vanishes as $T^2$, but finite impurity scattering would still contribute to the damping in the way we discussed earlier.  At finite temperatures, quasiparticle collisions would lead to Landau damping of the collective modes. \n\nThis simple calculation tells us that microscopically the interaction potential determines the dispersion of the electronic collective modes. The zero sound mode for short-range interactions becomes the gapped 3D plasmon mode in the presence of long-range Coulomb coupling. (The first sound mode also acquires a long wavelength gap as discussed earlier.) Indeed, from the perspective of symmetry, both sound modes and plasmons are density fluctuations that characterize the underlying particle number conservation. So they are actually the same collective mode, and it is not a surprise that they all develop long wavelength gaps because of the long-range Coulomb interaction. Thus, the hydrodynamic sound in 3D metals is not a sound mode at all since it has a finite energy at zero momentum defined by the 3D plasma frequency. We do note, however, that the sound modes differ from the plasmon in their next-to-leading-order dispersion corrections at finite momentum. \nAlthough the 2D case is clearly addressed in Ref.~[3],\nwe briefly discuss the RPA calculation in 2D for completeness. \nThe electron polarization function in two dimensions reads\n\\begin{eqnarray} \\label{eq:polarization2D}\n\t\\Pi({\\bf q}, \\omega) = \\frac{k_F^2 q^2}{4 \\pi m \\omega^2} \\left( 1 + \\frac34 \\frac{k_F^2}{m^2}\\frac{q^2}{\\omega^2} \\right).\n\\end{eqnarray}\nPutting the short-range potential given by $V({\\bf q}) = \\frac{2\\pi v_F}{ k_F} F_0$ and the polarization function~(\\ref{eq:polarization2D}) into the eigen equation~(\\ref{eq:RPA}), we recover the linear zero sound $\\omega = \\pm (\\frac{F_0}{2} + \\frac34 )^{1\/2}v_F q$.\nOn the other hand, using instead the 2D Coulomb potential given by $V_\\text{Cou}({\\bf q}) = \\frac12 \\frac{e^2}{|{\\bf q}|}$ (the factor of $\\frac12$ comes from the two dimensional Fourier transform of the usual Coulomb potential $V_\\text{Cou}({\\bf r}) = \\frac1{4\\pi} \\frac1{|\\bf r|}$) we obtain the following collective plasmon mode\n\\begin{eqnarray}\t\n\t\\omega = \\pm \\sqrt{\\frac{\\pi \\alpha v_F^2}{\\lambda_F} |\\bf q|}.\n\\end{eqnarray}\nTherefore, within the RPA calculation, one can already see that the linear sound wave is modified to be $\\omega \\propto \\sqrt{q}$ by replacing the short-range interaction by the long-range Coulomb interaction. Going beyond the RPA framework, it was shown in Ref.~[3]\nthat although both the first and second sound waves exhibit the same plasmon-like dispersion, the next-to-the-leading order momentum dependence differs.\n\nWith this in mind, we now briefly discuss the 1D case.  In 1D, the Boltzmann approach does not work simply because the Fermi liquid does not exist in the presence of any finite interaction~\\cite{Tomonaga:1950remarks, Luttinger:1963exactly}. \nThus, starting from the electron Hamiltonian including a single-particle dispersion (we take a parabolic dispersion for simplicity) and interaction potentials $V(\\bf q)$ like that given above is a good and simple way to look for sound or plasmon modes. \nThe electron polarization function is now given by~\\cite{Sarma:1985screening}\n\\begin{eqnarray}\n\t\\Pi(q, \\omega) = \\frac{m}{2\\pi q} \\ln \\left( \\frac{m^2 \\omega^2 - (k_F- \\frac{q}2)^2 q^2}{m^2 \\omega^2 - (k_F+ \\frac{q}2)^2 q^2} \\right).\n\\end{eqnarray}\nWe expect a sound wave-like (linear in momentum) mode when the interaction is short ranged. Indeed the short-range potential $V(q) = \\pi v_F F_0$ leads to the zero sound mode in 1D,\n\\begin{eqnarray}\n \t\\omega = \\pm v_F \\sqrt{F_0} q.\n\\end{eqnarray}\nHow does the Coulomb interaction affect this sound mode? The answer is simple, we just need to replace $V(q)$ with the 1D Coulomb potential~\\cite{Sarma:1985screening},\n\\begin{eqnarray}\n\tV_\\text{Cou}(q) = \\frac{e^2}{4\\pi} \\int dr \\frac{e^{\\mathfrak{i} q r}}{\\sqrt{r^2 + a^2}} = \\frac{e^2}{4\\pi} 2 K_0(a q),\n\\end{eqnarray}\nwhere $K_0(x)$ is the modified Bessel equation of the second kind and $a$ is a short-range cutoff introduced to make the integral converge in 1D ($a$ can be thought of as a lattice constant). Plugging the Coulomb potential and the polarization function in 1D into the eigen equation~(\\ref{eq:RPA}), the resultant long wavelength collective mode is \n\\begin{eqnarray}\n\t\\omega = \\pm \\frac{e}{\\pi} \\sqrt{\\frac{v_F}2} q \\sqrt{- \\ln \\frac{a q}2}, \\quad a q \\ll 1,\n\\end{eqnarray}\nwhich is nothing but the well-known plasmon mode in 1D. Although we consider zero temperature, we expect that the plasmon mode takes over the sound modes in 1D when the Coulomb interaction dominates since the dispersion is largely independent of temperature. A more physical argument is that from the symmetry perspective, sound modes and plasmons are the very same mode, and the different names just refer to whether the interaction potential is short-range or long-range. The curious thing is that in 3D systems, where the Coulomb coupling goes as $q^{-2}$, the sound mode is not acoustic at all since it acquires the plasmon mass at zero momentum.\n\n\nIn conclusion, we constructed a solvable model in 3D to obtain the sound modes in both the collisionless regime and the collision-dominated hydrodynamic regime. In particular, we discussed the effect of long-range Coulomb interaction on the sound modes. We found that in the presence of Coulomb interaction, both the zero sound and the first sound obtain a finite gap equal to the plasmon frequency, and a damping rate which is quadratic in momentum. We also discussed general long-range interactions that lead to unusual plasmon dispersions. Finally, we clarified the collective mode and sound mode dichotomy in 1D.\n\n\\section*{Acknowledgments}\n\nThis work is supported by the Laboratory for Physical Sciences. S.-K.J. is supported by the Simons Foundation via the It From Qubit Collaboration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nPrecise and reliable control of a quantum system is an attractive and\nchallenging experimental issue in quantum physics. \nIn particular, the importance of its application to quantum information\nprocessing has been increasing recently. \nA promising way to achieve this is to employ geometric\nphases (or, more generally, non-Abelian\nholonomies)\\,\\cite{Nakahara2003,ChruscinskiJamiolkowski2004}, because \ngeometric phases are expected to be robust against noise and decoherence\nunder a proper condition\\,\\cite{BlaisTremblay2003,Ota;Kondo:2009}. \nA large number of studies for applying their potential\nrobustness to quantum computing have been done, e,g., \nphase-shift gates with Berry phases\\,\\cite{jones2000}, \nnonadiabatic geometric quantum \ngates\\,\\cite{Wang;Keiji:2001,Zhu;Wang:2002,zhu2003,Tianetal2004,\nDing;Wang:2005,ZhuZanardi2005,Das;Kumar:2005,Imai;Moriaga:2007}, \nholonomic quantum computing\\,\\cite{Zanardi;Rasetti:1999,Duan;Zoller:2001,\nNiskanen;Salomaa:2003,Solinas;Zanghi:2004,Karimipour;Majd:2004,\nTanimura;Hayashi:2005,Goto;Ichimura:2007,Ota;Nakahara:2008}, \nquantum gates with noncyclic geometric\nphases\\,\\cite{Friedenauer;Sjoqvist:2003}, and so on.\n \nFor physical realization of geometric quantum gates, elimination\nof a dynamical phase is a key point. \nJones {\\it et al.}\\,\\cite{jones2000} implemented a controlled phase-shift\ngate with a Berry phase\\,\\cite{Berry,Simon:1987} \nby quasi-statistically, or slowly,  \ncontrolling an effective field in a rotating frame. \nThey nulled dynamical phase effects using a conventional  \nspin echo approach\\,\\cite{levitt}.  \nZhu and Wang\\,\\cite{zhu2003} proposed a geometric quantum gate based on \nthe Aharonov-Anandan phase\\,\\cite{AharonovAnandan1987,Page:1987},\nwhich should be fast since a nonadiabatic process is employed. \nIn their proposal, elimination of a dynamical phase is achieved \nby a double-loop method, where a dynamical phase cancels out while\na geometric phase accumulates along two loops. \n\nAlthough several experimental techniques for the application of\ngeometric phases to quantum computation are\navailable\\,\\cite{Tianetal2004,Das;Kumar:2005,Imai;Moriaga:2007}, explicit \nimplementations of geometric phase gates have not been extensively\nstudied so far.  \nWithout explicit implementations, the often-cited advantage of the\nholonomic quantum gates is nothing more than a desk plan. \nThus, such explicit examples are highly desirable. \nIn this paper, we combine Zhu and Wang's approach with Jones \n{\\it et al.}'s one, employing an Aharonov-Anandan phase for fast gate operation and a\nspin echo technique for dynamical phase cancellation,  \nand demonstrate one-qubit gates with a commercial liquid-state  \nnuclear magnetic resonance (NMR) system. \nIn many experiments of nonadiabatic geometric quantum\ngates\\,\\cite{Tianetal2004,Das;Kumar:2005,Imai;Moriaga:2007}, \nthe gate operations in which the dynamical phase is arranged to\nvanish\\,\\cite{Ota;Kondo:2009,SuterMuellerPines1988} have been adopted. \nIn the present paper, we show that we may have another option for\nphysical realization of geometric quantum gates. \n \nThe next section summarizes Zhu and Wang's theoretical proposal and  \nour modifications for practical implementation in liquid-state NMR \nquantum computer. \nIn Sec.\\,\\ref{sec:experiments} we describe details of our experiments,\nwhere implemented gates are evaluated by performing quantum process\ntomography. \nSec.\\,\\ref{sec:summary} is devoted to summary. \n \n\\section{Theory} \n\\label{sec:theory}\n\\subsection{Quantum gates with orthonormal cyclic vectors} \n\\label{subsec:qg_ocv}\nThe Aharonov-Anandan phase is a geometric phase associated with nonadiabatic  \ncyclic time evolution of a quantum \nsystem\\,\\cite{AharonovAnandan1987,Page:1987}.  \nLet us write a state at $t$ ($0\\le t\\le \\tau$) as $|{\\psi(t)}\\rangle$ \nin the Hilbert space $\\mathcal{H}$ with dimension $n$.  \nThe time evolution of a system is given by the Schr\\\"odinger equation \nwith a time-dependent Hamiltonian $H(t)$. \nWe take the natural unit in which $\\hbar=1$. \nThe nonadiabatic cyclic time evolution of the quantum system means that \n\\begin{equation*}\n|{\\psi(\\tau)}\\rangle = U(\\tau)|{\\psi(0)}\\rangle\n = e^{i\\gamma}|{\\psi(0)}\\rangle, \n\\end{equation*} \nwhere $U(\\tau)$ is the time evolution operator and $\\gamma \\in \\mathbb{R}$.  \nLet us call $|{\\psi(0)}\\rangle$ a cyclic vector\\,\\cite{AharonovAnandan1987}. \nWe note that the dynamical phase $\\gamma_{\\rm d}$ associated with this\ntime evolution is \n\\begin{equation} \n\\gamma_{\\rm d} = - \\int_0^\\tau \\langle{\\psi(t)}|H(t)|{\\psi(t)}\\rangle dt, \n\\label{eq:dyn_phase} \n\\end{equation} \nwhile the geometric component is defined as \n\\begin{equation*}\n \\gamma_{\\rm g} = \\gamma - \\gamma_{\\rm d}. \n\\end{equation*}\nWe take a set of orthonormal cyclic vectors, \n$\\{ |{u_k(0)}\\rangle \\}$ so that \n\\begin{equation*}\n|{u_k(\\tau)}\\rangle =  e^{i \\gamma_k} |{u_k(0)}\\rangle,\n\\end{equation*}\nwhere $k = 1,\\ldots,n$. \nHereafter, we write $|u_k(0)\\rangle$  as $|u_k\\rangle$ for brevity. \nA general state $|\\Psi(0) \\rangle\\in\\mathcal{H}$ is expressed as \n\\(\n |{\\Psi}(0) \\rangle = \\sum_{k=1}^{n}a_{k}|{u_{k}}\\rangle\n\\). \nThen, we have \n\\(\n|{\\Psi(\\tau)}\\rangle = U(\\tau)|\\Psi(0) \\rangle= \n\\sum_{k=1}^{n}a_{k}e^{i\\gamma_{k}}|u_{k}\\rangle\n\\). \nWe denote a fixed basis in $\\mathcal{H}$ as $\\{|l\\rangle \\}$,\nwhich corresponds to the computational basis $\\{|0\\rangle,|1\\rangle\\}$\nfor the case $n=2$. \nIn terms of $\\{|l\\rangle\\}$, we have \n\\(\n |\\Psi(\\tau)\\rangle = \\sum_{l=0}^{n-1}c_{l}(\\tau)|l\\rangle\n\\). \nIt means that \n\\(\n a_{k} \n=\n\\langle u_{k}|\\Psi(0)\\rangle \n= \n\\sum_{l=0}^{n-1}c_{l}\\langle u_{k}|l\\rangle\n\\), \nwhere $c_{l}=c_{l}(0)$. \nTherefore, we find that \n\\begin{equation} \nc_{l}(\\tau) \n= \n\\sum_{l^{\\prime}=0}^{n-1} \n\\sum_{k=1}^{n}\\,e^{i\\gamma_{k}} \n\\langle{l}|u_{k}\\rangle \\langle{u_{k}}|l^{\\prime} \\rangle c_{l^{\\prime}} \n= \n\\sum_{l^{\\prime}=0}^{n-1}\\,V_{ll^{\\prime}}c_{l^{\\prime}}, \n\\label{eq:coeff_tau_cb}\n\\end{equation} \nwhere  \n\\begin{equation} \nV_{l  l^{\\prime}} = \\sum_{k=1}^{n} \ne^{i\\gamma_{k}} \\langle{l }|u_{k}\n\\rangle \\langle{u_{k}}|l ^{\\prime} \\rangle. \n\\label{eq:top_cb}\n\\end{equation} \nWe have \n\\(\n  \\gamma_{k} = \\gamma_{k,{\\rm d}} + \\gamma_{k,{\\rm g}} \n\\), \nwhere $\\gamma_{k,{\\rm d}}$ and $\\gamma_{k,{\\rm g}}$ are the dynamical\nand the geometric phases associated with $|u_{k}\\rangle$, respectively. \n\n\\subsection{Single-qubit case} \nLet us consider a single-qubit system.  \nWe choose the Hamiltonian $H_1(t)$ as  \n\\begin{equation*} \nH_{1}(t) = -\\frac{1}{2} {\\bm \\Omega}(t)\\cdot\\vsigma, \n\\end{equation*} \nwith NMR in mind, where \n\\(\n\\vsigma =\n(\\sigma_{x},\\,\\sigma_{y},\\,\\sigma_{z})\n\\) \nand\n\\begin{equation} \n {\\bm \\Omega}(t)  =\n\\left( \n\\omega_{1} \\cos(\\omega_{\\rm rf} t-\\phi),-\n\\omega_{1}\\sin(\\omega_{\\rm rf} t- \\phi), \n\\omega_{0} \n\\right).\n\\label{eq:field} \n\\end{equation}\nWe note that ${\\bm \\Omega(t)}$ is a time dependent  \nparameter corresponding to the external field and  \n$\\sigma_{\\alpha}$ ($\\alpha=x,\\,y,\\,z$) is the $\\alpha$th component of the\nPauli matrices. \nOne can freely control $\\omega_0$ by\ntaking a proper rotating frame. \nThe transformation to the rotating frame with the frequency\n$\\omega_{\\rm{rf}}$ is made possible by the unitary transformation \n$U = e^{i \\omega_{\\rm{rf}} \\sigma_z t\/2}$ \nand the Hamiltonian in the rotating frame is \n\\begin{equation}\nH_{1{\\rm r}} \n=  U^{\\dagger} H_1 U - i U^{\\dagger}\n\\frac{d}{dt} U \n= -\\frac{1}{2} \\Omega \\vm \\cdot \\vsigma,\n\\label{eq:nmr_rot_ham}\n\\end{equation}\nwhere \n\\(\n\\Omega = (\\omega_{1}^{2}+\\Delta^{2})^{1\/2}\n\\), \n\\(\n\\Delta = \\omega_{0}-\\omega_{\\rm rf}\n\\), \n\\(\n\\vm =\n( \\sin\\chi\\cos\\phi,\\,\\sin\\chi\\sin\\phi,\\,\\cos\\chi )\n\\), and \n\\(\n\\tan\\chi = \\omega_{1}\/\\Delta \n\\). \nThe solution of the Sch\\\"odinger equation is \n\\begin{eqnarray} \n|\\psi (t)\\rangle \n&=& \ne^{i\\omega_{\\rm rf} t\\sigma_{z}\/2}\\, \ne^{i\\Omega t \\vm\\cdot\\vsigma\/2}\\, \n|\\psi (0)\\rangle .\n\\label{eq:sol_1q} \n\\end{eqnarray} \n\\begin{figure}[tbp]\n\\centering\n\\scalebox{0.58}[0.58]{\\includegraphics{fig1a}}\n\\scalebox{0.58}[0.58]{\\includegraphics{fig1b}}\n\\vspace{-3mm}\n\\flushleft\n(a)\\hspace{40mm}(b)\n\\caption{Example of dynamics of a single-qubit cyclic vector. \n(a) A time-dependent external field\n $\\bm{\\Omega}(t)$ and (b) a closed trajectory on the Bloch\n sphere corresponding to a cyclic vector $|\\psi_{+}(t)\\rangle$, \n$0\\le t\\le \\tau=2\\pi\/|\\omega_{\\rm rf}|$. \nThe end point of each arrow represents the initial value. \nWe set the parameters $\\omega_{0}=2\\pi$, $\\omega_{1}=0.5\\times 2\\pi$,\n $\\omega_{\\rm rf}=0.8\\times 2\\pi$, and $\\phi=0$ in\n Eq.\\,(\\ref{eq:field}). }\n\\label{fig:singlel}\n\\end{figure}\nWe denote the eigenstates of $\\vm\\cdot \\vsigma$ with eigenvalues \n$\\pm 1$ as \n\\(\n |\\psi_{\\pm}\\rangle\n\\). \nTheir explicit forms are  \n\\begin{eqnarray*} \n |{\\psi_{+}}\\rangle  \n&=&  e^{-i\\phi\/2} \\cos\\frac{\\chi}{2} |{0}\\rangle \n +   e^{i\\phi\/2}  \\sin\\frac{\\chi}{2} |{1}\\rangle ,  \\\\ \n |{\\psi_{-}}\\rangle  \n&=& - e^{-i\\phi\/2} \\sin\\frac{\\chi}{2} |{0} \\rangle\n +    e^{i\\phi\/2}  \\cos\\frac{\\chi}{2} |{1} \\rangle,  \n\\end{eqnarray*} \nwhere $|{0}\\rangle$ and $|{1}\\rangle$ are the eigenstates of $\\sigma_z$\nwith eigenvalues $+1$ and $-1$, respectively, and taken as the\ncomputational basis.  \nThe corresponding Bloch vectors of $|{\\psi_{\\pm}}\\rangle$ are  \n\\begin{equation*} \n\\langle{\\psi_{\\pm}}| \\vsigma |{\\psi_{\\pm}}\\rangle = \\pm \\vm .  \n\\end{equation*} \nWe require that $|{\\psi_{\\pm}}\\rangle$ be cyclic vectors. \nSince $|\\psi_{+}\\rangle$ and $|\\psi_{-}\\rangle$ are mutually orthogonal,\nthey are identified as $\\{|u_{k}\\rangle\\}$ in Sec.\\,\\ref{subsec:qg_ocv}.  \nIt follows from Eq.\\,(\\ref{eq:sol_1q}) that the execution time $\\tau$\nmust satisfy the condition $|\\omega_{\\rm rf}| \\tau\/2 = \\pi$, i.e., \n$\\tau = 2\\pi\/|\\omega_{\\rm rf}|$.\nThen, $|\\psi_{\\pm}(\\tau) \\rangle$ is written as\n\\begin{eqnarray*} \n|{\\psi_{\\pm}(\\tau)}\\rangle = e^{i \\gamma_{\\pm}} |{\\psi_{\\pm}}\\rangle, \n\\end{eqnarray*} \nwhere \n\\(\n\\gamma_{\\pm} = \\pi \\pm  \\pi \\Omega\/|\\omega_{\\rm rf}|\n\\). \nFigure \\ref{fig:singlel} shows an example of $\\bm{\\Omega}(t)$ and the\nclosed trajectory on the Bloch sphere corresponding to $|\\psi_{+}\\rangle$. \nUsing Eq.\\,(\\ref{eq:dyn_phase}), we find that the dynamical phase is \n\\begin{equation*} \n\\gamma_{\\rm \\pm, d}  \n= \\pm \\frac{\\tau}{2}(\\omega_1 \\sin \\chi + \\omega_0 \\cos \\chi)  \n= \\pm \\frac{\\pi(\\omega_1^2+\\omega_0 \\Delta)}{|\\omega_{\\rm rf}| \\Omega},\n\\end{equation*} \nwhile the geometric phase is \n\\begin{equation*} \n\\gamma_{\\pm,{\\rm g}}  \n= \n\\gamma_{\\pm} -\\gamma_{\\pm, {\\rm d}} \n=  \\pi \\left(1 \n\\mp \\frac{\\omega_{{\\rm rf}}}{|\\omega_{{\\rm rf}}|}\\frac{\\Delta}{\\Omega} \n\\right).\n\\end{equation*} \nBased on Eqs.\\,(\\ref{eq:coeff_tau_cb}) and (\\ref{eq:top_cb}), we obtain\na unitary gate with the dynamical and the Aharonov-Anandan phases with respect to the\ncomputational basis $\\{|0\\rangle,|1\\rangle\\}$, \n\\begin{equation*} \nV(\\tau)  \n=\n\\left( \n\\begin{array}{cc} \n\\cos\\gamma-i\\sin\\gamma\\cos\\chi \n&  - i e^{-i \\phi}\\sin \\gamma \\sin \\chi \\\\ \n- i e^{i \\phi}\\sin \\gamma \\sin \\chi \n& \\cos\\gamma +i\\sin\\gamma\\cos\\chi\n\\end{array} \n\\right), \n\\end{equation*} \nwhere $\\gamma = 2\\pi -\\gamma_{+}=\\gamma_{-}$ has been used to simplify\nthe notation.  \n\n\\subsection{Cancellation of dynamical phases} \n\\label{subsec:cancellation}\nWe closely follow Zhu and Wang's proposal\\,\\cite{zhu2003} \nin order to eliminate a dynamical phase. \nThey proposed the use of two successive unitary operations, in which \na dynamical phase cancels out while a geometric phase accumulates along\nthese two operations.  \nEach unitary operation associated with a loop is characterized by the\ntime-dependent external field (\\ref{eq:field}). \nThe loop parameter corresponding to the $i$th loop is denoted by\n$\\bm{\\Omega}_{i}(t)$ ($i=1,2$). \nThus, in principle, we have four independent parameters in each\nloop, i.e., $\\omega_{i,1}$, $\\omega_{i,{\\rm rf}}$, $\\phi_{i}$, and\n$\\omega_{i,0}$.  \nWe note that they are not always tunable in a real experimental\nsituation. \n\\begin{figure}[tbp]\n\\begin{center}\n\\scalebox{0.5}[0.5]{\\includegraphics{fig2}}\n\\end{center}\n\\caption{Schematic diagram of double-loop method for dynamical phase\n cancellation according to the proposal by Zhu and\n Wang\\,\\cite{zhu2003}. \nTwo time-dependent magnetic fields are applied sequentially. \nThe first magnetic field (loop 1) rotates counter-clockwise,\n while the second one (loop 2) rotates clockwise in order to eliminate a\n dynamical phase. \n\\label{fig:loops}}\n\\end{figure}\n\nWe will search for the condition under which\nthe dynamical phases associated with the two loops cancel each other as\nshown in Fig.\\,\\ref{fig:loops}. \nWe focus on the case in which $\\phi_{i}=0$ and \n\\(\n\\omega_{1,{\\rm rf}}=\\omega_{2,{\\rm rf}}(\\equiv \\omega_{\\rm rf}) >0\n\\) \nfor simplicity. \nThe first loop (loop 1) is described by\n\\begin{equation} \n\\bm{\\Omega}_1(t) \n=\n\\left( \n\\omega_{1,1} \\cos \\omega_{\\rm rf} t,\\, \n-\\omega_{1,1} \\sin \\omega_{\\rm rf} t,\\, \n\\omega_{1,0} \n\\right),\n\\label{loop_1}\n\\end{equation}\nwhile the second loop (loop 2) by\n\\begin{equation} \n\\bm{\\Omega}_2(t)\n= \n- \n\\left( \n\\omega_{2,1} \\cos \\omega_{\\rm rf} t,\\, \n-\\omega_{2,1} \\sin \\omega_{\\rm rf} t,\\, \n\\omega_{2,0} \n\\right) \\bm{R}_{y}(\\theta). \n\\label{loop_2}   \n\\end{equation} \nLet $\\bm{R}_{y}(\\theta) \\in {\\rm SO}(3)$ represent a rotation around the\n$y$-axis by an angle $\\theta$.  \nThe rotation angle $\\theta$ is chosen so that the corresponding cyclic \nvectors $|{\\psi_{i,\\pm}}\\rangle$ for these two loops satisfies \n\\begin{equation}\n|\\psi_{1,\\pm}\\rangle = e^{i c}|\\psi_{2,\\pm}\\rangle, \n\\label{eq:connect_cond}\n\\end{equation} \nwhere $c \\in \\mathbb{R}$. \nIn other words, the Bloch vectors corresponding to\n$|\\psi_{1,\\pm}\\rangle$ coincide with those to \n$|\\psi_{2,\\pm}\\rangle$. \nUsing the notation in Eq.\\,(\\ref{eq:nmr_rot_ham}), we find that in the\nloop $i$ \n\\begin{equation*} \n\\bm{m}_{i} \n=\n\\bm{k}_{i}\\bm{O}_{i},\n\\quad\n\\bm{k}_{i}\n= (\\sin\\chi_i,0,\\cos\\chi_i),\n\\end{equation*} \nwhere \n\\(\n\\bm{O}_{1} = \\bm{I}_{3}\n\\), \n\\(\n\\bm{O}_{2} = \\bm{R}_{y}(\\theta)\n\\), and we write \nthe $3 \\times 3$ unit matrix as\n\\(\n\\bm{I}_{3}\n\\). \nThe rotation angle $\\chi_{i}$ is defined as  \n\\begin{eqnarray}\n&&\n\\tan \\chi_1 = \\frac{\\omega_{1,1}}{\\Delta_1}, \\quad\n\\Delta_1 =\\omega_{1,0} - \\omega_{\\rm rf}, \n\\label{eq:def_del1_omega1}\\\\ \n&&\n\\tan \\chi_2 = \\frac{\\omega_{2,1}}{\\Delta_2}, \\quad \n\\Delta_2 = \\omega_{2,0} + \\omega_{\\rm rf}. \n\\label{eq:def_del2_omega2}\n\\end{eqnarray} \nThe angle $\\theta$ is explicitly given as \n\\begin{equation*}\n\\theta = \\chi_{2} - \\chi_{1}.\n\\end{equation*} \nFigure \\ref{fig:exampleZW}(a) shows an example of the time-dependent\nexternal fields ${\\bm \\Omega}_{1}(t)$ and ${\\bm \\Omega}_{2}(t)$. \nThe corresponding closed trajectory on the Bloch sphere is drawn in\nFig.\\,\\ref{fig:exampleZW}(b), in which the initial point corresponds to\na cyclic vector $|\\psi_{1,+}\\rangle$. \nFigure \\ref{fig:exampleZW}(b) shows that $|\\psi_{1,+}\\rangle$ is not\nonly cyclic for loop 1 but also for the total process (i.e., loop $1$ and\nloop $2$) due to the connection condition (\\ref{eq:connect_cond}). \n\\begin{figure}[tp]\n\\centering\n\\scalebox{0.59}[0.59]{\\includegraphics{fig3a}}\n\\scalebox{0.59}[0.59]{\\includegraphics{fig3b}} \n\\vspace{-3mm}\n\\flushleft\n(a)\\hspace{45mm}(b)\n\\caption{(color online) Example of Zhu-Wang's double-loop method. \nThe time-dependent external fields $\\bm{\\Omega}_{1}(t)$ and\n $\\bm{\\Omega}_{2}(t)$ are shown in (a), while the closed trajectory \non the Bloch sphere corresponding to the cyclic vectors\n $|\\psi_{1,+}\\rangle$ and $|\\psi_{2,+}\\rangle$ in (b). \nWe note that these are connected and thus form one closed\n trajectory. \nWe set the loop parameters $\\omega_{1,1}=\\omega_{2,1}=2\\pi$, $\\omega_{\\rm rf}=0.7\\times 2\\pi$,\n $\\omega_{1,0}=0.27\\times 2\\pi$, and $\\omega_{2,0}=1.5\\times 2\\pi$ in\n Eqs.\\,(\\ref{loop_1}) and (\\ref{loop_2}). We note that these parameters are\n calculated on the basis of a condition for nulling dynamical phases in\n Ref.\\,\\cite{zhu2003}. \nIn this example, $\\Gamma=\\frac{1}{2}$ in Eq.\\,(\\ref{eq:ori_geop}).}\n\\label{fig:exampleZW}\n\\end{figure}\nIt is necessary to search for $\\omega_{i,a}$ and $ \\omega_{\\rm rf}$ \n($a=0,1$) so that  \n\\begin{eqnarray}\n&&\n\\gamma_{\\rm 1,d} +\\gamma_{\\rm 2,d} = 0, \n\\label{eq:ori_no_dynp}\\\\ \n&&\n\\gamma_{\\rm 1,g} + \\gamma_{\\rm 2,g} = \\Gamma \\pi . \n\\label{eq:ori_geop}\n\\end{eqnarray} \nWe write them more explicitly as follows: \n\\begin{eqnarray} \n \\frac{\\omega_{1,1}^2 + \\omega_{1,0} \\Delta_1}\n      {\\Omega_1}\n&=&  \n\\frac{\\omega_{2,1}^2 + \\omega_{2,0} \\Delta_2}{\\Omega_2}, \n\\label{eq:con_d}\\\\ \n\\frac{\\Delta_1}{\\Omega_1}+\\frac{\\Delta_2}{\\Omega_2} \n&=&2-\\Gamma, \n\\label{eq:con_g}\n\\end{eqnarray} \nwhere $\\Omega_i = (\\omega_{i,1}^2+\\Delta_i^2)^{1\/2}$. \nThere may be many combinations of $\\omega_{i,a}$ and $\\omega_{\\rm rf}$\nfor a given $\\Gamma$ which satisfy the conditions (\\ref{eq:con_d}) and \n(\\ref{eq:con_g})\\,\\cite{zhu2003}. \nWe note that a set of the parameters employed in\nFig.\\,\\ref{fig:exampleZW} is one example for the solution of\nEqs.\\,(\\ref{eq:con_d}) and (\\ref{eq:con_g}), in which\n$\\Gamma=\\frac{1}{2}$. \n\nAfter the elimination of a dynamical phase, we have a one-qubit\ngeometric quantum gate \n\\begin{equation}\n V_{\\rm ZW} \n= e^{i\\Gamma\\pi}|\\psi_{1,+}\\rangle\\langle \\psi_{1,+}|\n+\ne^{-i\\Gamma\\pi}|\\psi_{1,-}\\rangle\\langle \\psi_{1,-}|. \n\\label{eq:ZW_gate}\n\\end{equation}\n\n\\subsection{Spin-echo approach }\n\\label{subsec:spin_echo}\nZhu and Wang's proposal for eliminating a dynamical phase is not\nfeasible for a conventional commercial NMR system where a field along\nthe $z$-axis is strictly constant. \nIn other words, it is difficult to realize $\\bm{\\Omega}_2(t)$ in\nEq.\\,(\\ref{loop_2}). \n\\begin{figure}[tp]\n\\begin{center}\n\\scalebox{0.45}[0.45]{\\includegraphics{fig4}}\n\\end{center}\n\\caption{Schematic diagram of double-loop method for dynamical phase\n cancellation on the basis of a spin-echo approach.  \nTwo (four) soft (hard) square pulses are applied. \nWe note that $R_{y}(\\theta)=e^{-i\\theta\\sigma_{y}\/2}$, in which\n $\\theta=\\chi_{2}-\\chi_{1}$.  \n\\label{fig:pseqloop}}\n\\end{figure}\nIn the present paper, we propose an experimentally feasible method, in\nwhich the loop 2 is divided into three successive steps while the loop 1\nremains unchanged. \nThe separation of the loop 2 is motivated by the spin-echo technique\nfrequently employed in NMR experiments, in which the direction of time\nis reversed by an application of a pair of $\\pi$-pulses. \nThree successive operations are \n(a) a rotation around the $y$-axis by \n\\(\n\\theta (= \\chi_{2}-\\chi_{1})\n\\), \n(b) an operation corresponding to precession by a field \n\\(\n-\\left(\n\\omega_{2,1} \\cos \\omega_{\\rm rf}t,\\,\n\\omega_{2,1} \\sin \\omega_{\\rm rf}t,\\, \n\\omega_{2,0} \\right)\n\\)\nfor a duration $\\tau = 2\\pi\/|\\omega_{\\rm rf}|$, \nand (c) a rotation around the $y$-axis by $-\\theta$.  \nThe rotations $R_{y}(\\pm \\theta)$ correspond to the basis\nvector change and back as shown in Fig.\\,\\ref{fig:pseqloop}. \nRotation around the $y$-axis by $\\theta$ is easy to realize by\na radio-frequency (rf) pulse, which corresponds to the unitary operator\n\\begin{equation}\nR_{y}(\\theta) = e^{-i \\theta \\sigma_y\/2}. \n\\end{equation}\nWe emphasize here that \n\\(\n|\\psi_{2,\\pm}\\rangle = R_{y}(\\theta)|\\psi_{1,\\pm}\\rangle \n\\).   \nWe assume that the pulse duration is infinitely short for simplicity. \nIt should be noted that this operation does not generate a dynamical phase\nsince the $y$-axis is perpendicular to both $\\bm{k}_1$ and \n$\\bm{k}_2$\\,\\cite{Ota;Kondo:2009,SuterMuellerPines1988}. \nThe operation corresponds to a precession \nby a field \n\\(\n-\\left(\n\\omega_{2,1}\\cos\\omega_{\\rm rf}t,\\, \n\\omega_{2,1}\\sin\\omega_{\\rm rf}t,\\, \n\\omega_{2,0} \\right)\n\\)\nfor $\\tau = 2\\pi\/|\\omega_{\\rm  rf}|$ is given as \n\\begin{equation*} \nU_{2}^{\\prime} =\ne^{i\\Omega_{2} \\tau (-\\bm{k}_2) \\cdot\\vsigma\/2}, \n\\end{equation*} \nfrom Eq.\\,(\\ref{eq:sol_1q}). \nThe identity \n\\begin{equation*}\nR_{y}(-\\pi) e^{i\\theta \\bm{k}_{2} \\cdot\\vsigma\/2} \nR_{y}(\\pi) =\ne^{i\\theta (-\\bm{k}_{2}) \\cdot\\vsigma\/2}, \n\\end{equation*} \nimplies that $U_2^{\\prime}$ can be realized by a precession\nunder the field \n\\(\n \\left(\n\\omega_{2,1} \\cos\\omega_{\\rm rf}t,\\, \n-\\omega_{2,1} \\sin\\omega_{\\rm  rf}t,\\,\n\\omega_{2,0} \\right)\n\\)\nfor $\\tau = 2\\pi\/|\\omega_{\\rm rf}|$ sandwiched by a pair of \n$\\pm\\pi$-pulses\\,\\cite{levitt}. \nWe again assume that $R_{y}(\\pm\\pi)$ is implemented for an infinitely\nshort pulse for simplicity. \n\nSummarizing the above arguments, the total process is described by \n\\(\nR_{y}(-\\theta)R_{y}(-\\pi)U_{2}(\\tau)R_{y}(\\pi)R_{y}(\\theta)U_{1}(\\tau)\n\\)\\,\\cite{comment1}, which is equivalent to Eq.\\,(\\ref{eq:ZW_gate}) if\nthe dynamical phase is zero. \nThe unitary operator $U_{i}(\\tau)$ is defined as \n\\(\nU_{i}(\\tau) \n= \ne^{i\\omega_{\\rm rf}\\tau\\sigma_{y} \/2}\ne^{i\\Omega_{i}\\bm{k}_{i}\\cdot\\vsigma \/2}\n\\).  \nThe geometric gate which we are going to demonstrate takes the form \n\\begin{equation}\nU_{\\rm echo} \n= R_{y}(-\\pi)U_{2}(\\tau)R_{y}(\\pi)R_{y}(\\theta)U_{1}(\\tau). \n\\label{eq:gate_by_pulse}\n\\end{equation}\nAlthough the complete realization of Zhu and Wang's original proposal\n(\\ref{eq:ZW_gate}) requires $R_{y}(-\\theta)$ at the end of the process\n(\\ref{eq:gate_by_pulse}), we can omit it for constructing a geometric quantum gate since\n$R_{y}(-\\theta)$ does not generate any dynamical phase here. \nWe note that \n\\(\nV_{\\rm ZW} \n= R_{y}(-\\theta) U_{\\rm echo}\n\\). \nUnder the conditions (\\ref{eq:ori_no_dynp}) and (\\ref{eq:ori_geop}), the\nmatrix representation of Eq.\\,(\\ref{eq:gate_by_pulse}) in the\ncomputational basis $\\{|0\\rangle,|1\\rangle\\}$ is given by  \n\\begin{widetext}\n\\begin{equation}\nU_{\\rm echo}\n=\n\\left(\n\\begin{array}{cc}\n\\cos(\\Gamma \\pi)\\cos(\\theta\/2)-i\\sin(\\Gamma \\pi)\\cos\\Theta \n& \n-\\cos(\\Gamma \\pi)\\sin(\\theta\/2)-i\\sin(\\Gamma \\pi)\\sin\\Theta \n\\\\\n\\cos(\\Gamma \\pi)\\sin(\\theta\/2)-i\\sin(\\Gamma \\pi)\\sin\\Theta \n&\n\\cos(\\Gamma \\pi)\\cos(\\theta\/2)+i\\sin(\\Gamma \\pi)\\cos\\Theta \n\\end{array}\n\\right), \n\\label{eq:mrep_gate_pulse}\n\\end{equation}\n\\end{widetext} \nwhere \n\\(\n\\Theta = (\\chi_{2}+\\chi_{1})\/2\n\\). \nWe note that $\\Gamma\\pi$ is the total geometric phase. \nThe pulse sequence (\\ref{eq:gate_by_pulse}) leads to intuitive\nunderstanding of the cancellation mechanism of the dynamical phase. \nLet us consider the case of $\\chi_{1}=\\chi_{2}$, i.e., the two loop are\ncompletely identical, for simplicity. \nIf no $\\pi$-pulse is applied, the dynamical property of the loop 1 is\nthe same as that of the loop 2 and the total dynamical phase is the addition\nbetween the contributions from the loops 1 and 2. \nIt should be noted here that the $\\pm \\pi$-pulses induce the time-reversal\ndynamics in the form of $U^{\\prime}_{2}$ in the loop 2. \nUnder the time-reversal transformation, the sign of the dynamical phase\nassociated with the loop 2 is inverted and hence the dynamical phase is\ncompletely eliminated. \nIt is necessary to employ different processes between the loops 1 and 2 to\nprevent the cancellation of the geometric phase associated with the two loops.  \nThe matrix representation (\\ref{eq:mrep_gate_pulse}) implies that \n$U_{\\rm echo}$ contains three parameters $\\Gamma$, $\\theta$, and\n$\\Theta$. \nDue to the limitation in the control parameters, it may be\ndifficult to choose them independently in a standard liquid-state NMR. \nWe will show that $\\theta$ and $\\Theta$ should be regarded as functions \nof $\\Gamma$\nand $\\omega_{1}$ in Sec.\\,\\ref{subsec:implementation}, in order to\nsatisfy Eqs.\\,(\\ref{eq:ori_no_dynp}) and (\\ref{eq:ori_geop}) as shown in\nTable \\ref{table:sol}.  \nOn the other hand, we are still able to use the rf phase $\\phi$. \nThus, we have the necessary number of free parameters to express\narbitrary elements of SU(2). \n\n\\subsection{Implementation in liquid-state NMR}\n\\label{subsec:implementation}\nWe implement the double-loop scheme in liquid-state NMR.\nWe take different loop parameterization from that of Zhu and Wang for\nease of implementation.  \nWe consider the system in a rotating frame defined by $\\omega_{\\rm ref}$. \nHereafter, we will denote an angular frequency $x$ measured with respect\nto this rotating frame as $\\tilde{x}\\equiv x-\\omega_{\\rm ref}$. \nThus, one can explicitly understand which quantities are controllable by\nchoosing a proper rotating frame. \n\nWe take a common value \n\\begin{eqnarray*}\n\\tilde{\\omega}_{0}=\\omega_{0}-\\omega_{\\rm ref}< 0\n\\end{eqnarray*}\nto $\\omega_{1,0}$ and $\\omega_{2,0}$. \nThe value of $\\tilde{\\omega}_{0}$ in the experiment will be explained in\nSec.\\,\\ref{subsec:pulse_seq}.  \nSimilarly, we assume that $\\omega_{1,1}=\\omega_{1}$ and\n$\\omega_{2,1}=\\omega_{1}$. \nInstead of these simplification, we allow different values with respect\nto $\\omega_{{\\rm rf}}$ between the two\nloops, i.e. $\\omega_{\\rm rf}=\\omega_{1,{\\rm rf}}$ in \nEq.\\,(\\ref{loop_1}) and $\\omega_{\\rm rf}=\\omega_{2,{\\rm rf}}$ in\nEq.\\,(\\ref{loop_2}). \nThese changes do not alter\nthe basic strategy for searching parameters that satisfy \n$\\gamma_{\\rm 1,d} + \\gamma_{\\rm 2,d}=0$\nand \n$\\gamma_{\\rm 1,g} + \\gamma_{\\rm 2,g}= \\Gamma \\pi$. \nWe consider the two loops in the rotating frame in which the frequency \nis $\\tilde{\\omega}_{i,{\\rm rf}}$, the amplitude $\\omega_{1}$,\nand the duration $\\tau_i=2 \\pi \/|\\tilde{\\omega}_{i,{\\rm rf}}|$, i.e.\n\\begin{equation*}\n\\tilde{{\\bm \\Omega}}_i(t) = \n(\\omega_1 \\cos \\tilde{\\omega}_{i, \\rm rf}\\,t, \n-\\omega_1 \\sin \\tilde{\\omega}_{i, \\rm rf}\\,t,\n\\tilde{\\omega}_0)\n\\quad\n(0\\le t\\le \\tau_{i}).\n\\end{equation*}\nThe solutions $\\tilde{\\omega}_{i,{\\rm rf}}\/|\\tilde{\\omega}_0|$ are numerically\nobtained for given $\\epsilon(\\equiv \\omega_1\/|\\tilde{\\omega}_0|)$ and\n$\\Gamma$. \n\n\\begin{table}[bp]\n\\begin{tabular}{ccccccccc}\n\\hline \\hline\n$\\epsilon$ \n&& $\\tilde{\\omega}_{1, \\rm rf}\/\\tilde{\\omega}_0$ \n&& $\\tilde{\\omega}_{2, \\rm rf}\/\\tilde{\\omega}_0$ \n&& $\\theta$~[rad] && $\\Theta $~[rad]\\\\\n\\hline \\hline\n0.5 && -0.6815 && 0.7803 \n    && -0.7298  && -0.639 \\\\\n0.3 && -0.8221 && 1.105  \n    && -0.9571  && -0.589 \\\\\n0.1 && -0.9422 && 1.609  \n    && -1.008   && -0.542 \\\\ \n\\hline\n\\end{tabular}\n\\caption{\n\\label{table:sol}\nSolutions of Eqs.\\,(\\ref{eq:ori_no_dynp}) and (\\ref{eq:ori_geop}) for\n $\\Gamma = \\frac{1}{2}$ and \n$\\epsilon =\\omega_1\/|\\tilde{\\omega}_0| = 0.5,\\,0.3,\\,0.1$ in the rotating\n frame so that $\\tilde{\\omega}_{0}<0$. \nWe note that \n$ \\tau_i |\\tilde{\\omega}_{i, \\rm rf}| =  2\\pi$, \n$\\theta=\\chi_{2}-\\chi_{1}$, \nand \n$\\Theta=(\\chi_{1}+\\chi_{2})\/2$.}\n\\end{table}\n\nWe summarize our parameter choice. \nFirst of all, we adopt a common value to $\\omega_{1,0}$ and\n$\\omega_{2,0}$, i.e., $\\tilde{\\omega}_{0}$ in the rotating frame defined\nas the angular frequency $\\omega_{\\rm ref}$. \nThe value of $\\omega_{1}(=\\omega_{1,1}=\\omega_{1,2})$ is given by \n\\(\n\\omega_{1} = \\epsilon |\\tilde{\\omega}_{0}|\n\\), in which $\\epsilon$ is a positive number. \nFor a given $\\epsilon$ and an aimed geometric phase $\\Gamma$, we can\nnumerically find proper $\\tilde{\\omega}_{i,{\\rm rf}}$ so that \n\\(\n\\gamma_{1,{\\rm d}}+\\gamma_{2,{\\rm d}}=0\n\\) and \n\\(\n\\gamma_{1,{\\rm g}} + \\gamma_{2,{\\rm g}}=\\Gamma \\pi\n\\). \nThe results for $\\Gamma=\\frac{1}{2}$ and $\\epsilon =0.5$, $0.3$, and 0.1,\nfor example, are shown in Table \\ref{table:sol}. \nFrom the observation of Eqs.\\,(\\ref{eq:def_del1_omega1}) and\n(\\ref{eq:def_del2_omega2}), the sign of $\\tilde{\\omega}_{1, \\rm rf}$\nshould be opposite to the one of $\\tilde{\\omega}_{2, \\rm rf}$. \nIt should be noted that the parameters given in Table \\ref{table:sol} are\ncompatible with this requirement. \nThe resultant geometric quantum gate is\nEq.\\,(\\ref{eq:mrep_gate_pulse}). \nThe values of $\\theta(=\\chi_{2}-\\chi_{1})$ and\n$\\Theta(=(\\chi_{2}+\\chi_{1})\/2)$ are given in Table \\ref{table:sol}. \n\nWhen $\\Gamma=\\frac{1}{2}$, Eq.\\,(\\ref{eq:mrep_gate_pulse}) takes the form \n\\begin{eqnarray}\nU_{\\rm echo}(\\Theta)\n&=& \ne^{-i \\pi\\sigma_y\/2}\ne^{i\\Omega_2 \\tau_2 \\bm{k}_2 \\cdot\\vsigma\/2} \\nonumber \\\\\n&&\n\\quad\n\\times \ne^{-i (\\theta+\\pi)\\sigma_y\/2}\ne^{i\\Omega_1 \\tau_1 \\bm{k}_1 \\cdot\\vsigma\/2} \\nonumber \\\\\n&=& e^{-i \\pi\/2}\n\\left( \\begin{array}{cr}\n          \\cos \\Theta &  \\sin \\Theta \\\\\n          \\sin \\Theta & -\\cos \\Theta\n       \\end{array}\n\\right),\n\\label{eq:1q_gate}\n\\end{eqnarray} \nwhich we experimentally demonstrate in the next section.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\subsection{Sample and spectrometer}\nWe implement a one-qubit gate described by Eq.\\,(\\ref{eq:1q_gate})\nwith a conventional commercial NMR system. \nWe employed a JEOL ECA-500 NMR spectrometer\\,\\cite{jeol} whose \nhydrogen Larmor frequency is approximately 500\\,MHz.\n$^{13}$C nucleus in a 0.6\\,ml, 0.2\\,M sample of ${}^{13}$C-labeled \nchloroform (Cambridge Isotope) in d-6 acetone is employed \nas a qubit, while protons are decoupled by a standard decoupling \ntechnique, called WALTZ\\,\\cite{levitt}. \nWe have chosen $^{13}$C-labeled chloroform for future experiments\ninvolving two-qubit gates. \nThe transverse and the longitudinal relaxation times are\n$T_2 \\sim 0.3\\,{\\rm s}$ and $ T_1 \\sim 5\\,{\\rm s}$, respectively.\nThe longitudinal relaxation time is shorten by adding a small amount of \nIron(I\\!I\\!I)-acetylacetonate so that a repetition rate \ncan be increased. $T_2$ and $T_1$ without Iron(I\\!I\\!I)-acetylacetonate\nare $\\sim 0.3\\,{\\rm s}$ and $\\sim 20\\,{\\rm s}$, respectively. \n\n\\subsection{Pulse sequence}\n\\label{subsec:pulse_seq}\nAs we discussed in the previous section, the gate (\\ref{eq:1q_gate}) \ncan be realized with two rotating magnetic fields and two hard \n(short) pulses. \nThe rotating fields are effectively obtained by two soft (long) pulses\nwhich are rotating with different frequencies \n\\mbox{$\\tilde{\\omega}_{i,\\rm rf} = \\omega_{i, \\rm rf} - \\omega_{\\rm ref}$} \n($i = 1, 2$) in the rotating frame with frequency $\\omega_{\\rm ref}$. \nThe first soft pulse (loop 1) is a usual square\npulse, while the second soft pulse (loop 2) is a\n(frequency) shifted laminar square pulse (SLP)\\,\\cite{ECA500}. \nThis SLP is employed in order to obtain the same phase $\\phi$ in\nEq.\\,(\\ref{eq:field}) for loop 2 as that for loop 1, i.e.,\n$\\phi_{1}=\\phi_{2}$.  \n\nWe take $|\\tilde{\\omega}_0| = 2 \\pi \\times 1000\\,{\\rm rad}\/{\\rm s}$\nand $\\phi_{i} = 0$ throughout the experiments. \nThe condition $\\phi_{i} = 0$ is taken for \nsimplicity as mentioned in the beginning of \nSec.\\,\\ref{subsec:cancellation}.\nWe independently calibrate the strengths of the soft and\nhard pulses in order to minimize a non-linearity error in setting the\nrf pulse amplitude. \nThe duration $t_{\\rm hp}$ of a hard $ \\pi $-pulse is set to \n$ 21.6\\,\\mu{\\rm s}$ throughout the experiments. \nWe ignore $t_{\\rm hp}$ in setting the phase of the second soft pulse, \nwhich is justified by the fact that \n$t_{\\rm hp} |\\tilde{\\omega}_0| \\ll 2\\pi$. \nThe precision of pulse duration control is 100\\,ns. \nThe durations $t_{i, \\rm sp}$ of two soft pulses are set to\n\\begin{eqnarray*}\nt_{i, \\rm sp} |\\tilde{\\omega}_{i, \\rm rf}|= 2 \\pi. \n\\end{eqnarray*}\nWe demonstrate three different gates with $\\epsilon=0.5,0.3$ and $0.1$. \nWe note that the phase of the second hard pulse \ncorresponding to $R_{y}(-\\pi)$ must be adjusted, \npresumably because the oscillator in the NMR spectrometer is \ndisturbed in generating a SLP. \nIt should be recalled that a SLP employs intensive phase modulation. \n\n\\subsection{Results}\nImplemented gates with $\\Gamma = \\frac{1}{2}$ are evaluated by performing  \nquantum process tomography\\,\\cite{qpt}. \nThe practical details are explained in Ref.\\,\\cite{kondo}. \nA quantum process ${\\mathcal E}$, such as a gate operation or relaxation\nprocess, is \n\\begin{equation*}\n\\rho \\mapsto \n{\\mathcal E}(\\rho) \n= \\frac{\\sum_{k} E_{k} \\rho E_{k}^\\dagger}{\\sum_{k}E^{\\dagger}_{k}E_{k}}\n\\end{equation*}  \nin the operator sum (or Kraus) representation\\,\\cite{Kraus:1983,op_sum_r}. \nWhen all $E_{k}$'s are determined, $\\mathcal{E}$ is considered \nto be identified. \nThis identification is called quantum process tomography. \n\n\\begin{figure}[tbp]\n\\centering\n\\scalebox{0.21}[0.21]{\\includegraphics{fig5a}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(a)\n\\end{flushleft} \n\\scalebox{0.21}[0.21]{\\includegraphics{fig5b}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(b)\n\\end{flushleft} \n\\scalebox{0.21}[0.21]{\\includegraphics{fig5c}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(c)\n\\end{flushleft} \n\\scalebox{0.21}[0.21]{\\includegraphics{fig5d}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(d)\n\\end{flushleft} \n\\caption{\n\\label{fig:results}\nGate operations visualized. The Bloch sphere \nin (a) is mapped to the surfaces in (b), (c), and (d) under the gates with\n $\\epsilon = 0.5, 0.3$ and $0.1$, respectively. \nThe right surface in (a) is an expected Bloch sphere when \n$\\Theta =-\\pi\/4$, which corresponds to the Hadamard gate. \nEach left panel in (b), (c), and (d) corresponds to the theoretical\n final state. \nThe middle panels are the results for the single gate operation \n$U_{\\rm echo}(\\Theta)$. \nThe right panels are for the two-successive (double) gate\n operations. }\n\\end{figure}\n\n\\begin{table}[tbp]\n\\begin{tabular}{c|cc|cccc}\n\\hline \\hline\n  $\\epsilon$ \n& $F_{\\rm e}(I_{0}, {\\mathcal E}\\circ\\mathcal{U}_{\\rm echo}^{-1})$ \n& ${\\rm Tr}[{\\mathcal E}(I_{0})] $  &\n& $F_{\\rm e}(I_{0}, {\\mathcal E}^2)$ \n& ${\\rm Tr}[{\\mathcal E}^2(I_{0})] $ \\\\\n\\hline \\hline\n0.5 & 0.75 & 1.00 && 0.74 &  1.02 \\\\\n0.3 & 0.88 & 1.08 && 0.83 &  1.07 \\\\\n0.1 & 0.84 & 1.07 && 0.85 &  1.06 \\\\\\hline\n\\end{tabular}\n\\caption{\n\\label{tab:entangle}\nThe entanglement fidelities for single and double operations \nwith $\\epsilon = 0.5$, $0.3$, and $0.1$. }\n\\end{table}\nThe Bloch sphere in Fig.\\,\\ref{fig:results}(a) is mapped under the gate\noperations to the surfaces in Figs.~\\ref{fig:results}(b), (c) and\n(d), which correspond to $\\epsilon = 0.5, 0.3$ and $0.1$, respectively. \nIf the gate operations are perfect, the surfaces are the spheres of unit\nradius (i.e., the Bloch sphere). \nThe left panel of each row shows the theoretical final state. \nIn the middle panels, the results for the single gate operation are\nshown. \nFinally, the right panels are for the two-successive (double) gate\noperation. \nThe Hadamard gate obtained when $\\Theta = -\\pi\/4$ is, for comparison, shown \nin the right pannel of Fig.\\,\\ref{fig:results}(a). \nFrom these figures, we find that $U_{\\rm echo}(\\Theta)$ \nin Eq.\\,(\\ref{eq:1q_gate}) is implemented although it is not perfect.  \n\nWe numerically evaluated the fidelity of the implemented gate using the\nentanglement fidelity\\,\\cite{kondo,op_sum_r} given by \n\\begin{eqnarray*}\nF_{\\rm e}(I_{0},\\mathcal{E}\\circ\\mathcal{U}_{\\rm echo}^{-1})\n&=&\n\\frac{\\sum_{k}|{\\rm Tr}[E_{k}U_{\\rm echo}(-\\Theta)I_{0}]|^2}\n{{\\rm Tr}[\\sum_k E_k U_{\\rm echo}\n(-\\Theta)I_0 U_{\\rm echo}^\\dagger(-\\Theta) E_k^\\dagger ]} \\nonumber \\\\\n&=&\n\\frac{\\sum_{k}|{\\rm Tr}[E_{k}U_{\\rm echo}(-\\Theta)I_{0}]|^2}\n{{\\rm Tr}(\\sum_k E_k I_0 E_k^\\dagger )},\n\\end{eqnarray*}\nwhere $\\mathcal{U}_{\\rm echo}$ is a super operator corresponding to the\nunitary operator $U_{\\rm echo}(\\Theta)$ (i.e., \n\\(\n\\mathcal{U}_{\\rm echo}(\\rho) = U_{\\rm echo}(\\Theta)\\rho U^{\\dagger}_{\\rm echo}(\\Theta)\n\\)),  \n$I_{0} = \\openone\/2$, and $\\openone$ is the identity matrix of dimension $2$. \nOne can find that \n\\(\nF_{\\rm e}(I_{0}, {\\mathcal E}\\circ\\mathcal{U}_{\\rm echo}^{-1}) = 1\n\\)\nwhen the gate operation $\\mathcal{E}$ is perfect.  \nIn the case of two successive gate \noperation, $F_{\\rm e}(I_{0}, {\\mathcal E}^2)$ gives a measure of \nthe fidelity since $[U_{\\rm echo}(\\Theta)]^{2} = -\\openone$. \nThe entanglement fidelities corresponding to the gate operations \nare summarized in Table \\ref{tab:entangle}. \n\nThe fidelities of the demonstrated gates are not high. \nThis may be attributed to the inhomogeneous rf field. \nThe free induction decay signal of the thermal state \nafter a $5\\pi\/2$-pulse, which corresponds to the operation \n\\(\ne^{-5\\pi\\sigma_{x}\/4}\n\\) for example, reduces to about $85$\\% of that after \na $\\pi\/2$-pulse, which corresponds to the operation \n\\(\ne^{-\\pi\\sigma_{x}\/4} \n\\) for example. \nThis fact indicates that there is some\nrf field inhomogeneity which may account for most of the reduction in \nthe fidelities in Table \\ref{tab:entangle}. \nPulse sequences in usual NMR operations are designed so that the \nrf field inhomogeneity does not affect measurements, for example,\nby employing composite pulses. Such techniques are not available \nin our experiments. \n\n\\section{Summary}\n\\label{sec:summary}\nWe demonstrated the elimination of the dynamical phase and\nthe implementation of the quantum gates with pure nonadiabatic\ngeometric phases in a liquid-state NMR quantum computer, based on the\ndouble-loop method.  \nBy means of a spin echo technique, we modified the original proposal so\nthat quantum gates are implemented in a standard high precession NMR\nsystem for chemical analysis. \nWe have proposed and experimentally verified an alternative method to eliminate\ndynamical phase. \nThe extension of the present method to two-qubit operations is an\nimportant future work\\,\\cite{comment2}. \nWe believe that our work is the first step toward physical realization of\nworking geometric quantum gates and further efforts should be made for\nimprovement of the gates. \n\n\\begin{acknowledgments} \nThis work was supported by ``Open Research Center'' Project for \n Private Universities: Matching fund subsidy from MEXT (Ministry of \n Education, Culture, Sports, Science and Technology).  \nMN's work is supported in part by Grant-in-Aid for Scientific Research\n (C) from JSPS (Grant No. 19540422).  \n\\end{acknowledgments} \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe complexity of the constraint satisfaction problem (CSP) with a fixed target\nstructure is a well established field of study in combinatorics and computer\nscience (see~\\cite{nesetril} for an overview). In the last\ndecade, we have seen algebraic tools brought to bear on the question of CSP\ncomplexity, yielding major new results (see e.g.~\\cite{BJK},\n\\cite{larossetesson-introduction}, \\cite{libor}). \n\nIn the algebraic approach, it is customary to study relational structures that\ncontain all possible constants. If ${\\mathbb A}$ is such a structure and we are to\ndecide the existence of a homomorphism $f:{\\mathbb B}\\to {\\mathbb A}$ then the constant\nconstraints prescribe values for $f$ at some vertices of ${\\mathbb B}$. We are thus\ndeciding if some partial homomorphism\n$f_c:{\\mathbb B} \\to {\\mathbb A}$ can be extended to the whole ${\\mathbb B}$. Therefore, $\\operatorname{CSP}({\\mathbb A})$\nbecomes the \\emph{homomorphism extension problem} with target structure ${\\mathbb A}$,\ndenoted by $\\operatorname{EXT}({\\mathbb A})$. It is easy to see that $\\operatorname{CSP}({\\mathbb A})$ reduces to $\\operatorname{EXT}({\\mathbb A})$,\nsince in CSP we extend the empty partial homomorphism.\n\nIn \\cite{nesetril-random}, the authors prove that $\\operatorname{CSP}({\\mathbb A})$ is almost surely\nNP-complete for ${\\mathbb A}$ large random relational structure with at least one at\nleast binary relation and without loops. We show by a different method that\nthe same hardness result holds for $\\operatorname{EXT}({\\mathbb A})$ even if we allow loops.\n\n\n\\section{Preliminaries}\nA \\emph{relational structure} ${\\mathbb A}$ is any set $A$ together with a family of\nrelations $\\{R_i:i\\in I\\}$ where $R_i\\subset A^{n_i}$. We call the number\n$n_i$ the \\emph{arity} of $R_i$. The sequence $(n_i:i\\in I)$ determines the\n\\emph{similarity type} of ${\\mathbb A}$. We consider only finite structures (and\nfinitary relations) in this paper. We use the notation $[n]=\\{1,2,\\dots,n\\}$. \n\nLet ${\\mathbb A}=(A,\\{R_i:i\\in I\\})$ and ${\\mathbb B}=(B,\\{S_i:i\\in I\\})$ be two relational structures of the same\nsimilarity type. A mapping $f: A\\to B$ is a homomorphism if for every $i\\in I$\nand every $(a_1,\\dots,a_{n_i})\\in R_i$ we have $(f(a_1),\\dots,f(a_{n_i}))\\in\nS_i$. \n\nLet us fix some $p\\in(0,1)$ and let $A$ be a set. The relation $S\\subset A^l$\nis an \\emph{$l$-ary random relation} on $A$ if every possible $l$-tuple belongs to $S$\nwith probability $p$ (independently of other $l$-tuples). We will call any\nrelational structure with one or more random relations a \\emph{random\nrelational structure}. In particular, a random relational structure with just\none binary relation is a \\emph{random digraph}.\n\nThe \\emph{Constraint Satisfaction Problem} with the target structure ${\\mathbb A}$,\ndenoted by $\\operatorname{CSP}({\\mathbb A})$, consists of deciding whether a given input relational\nstructure ${\\mathbb B}$ of the same similarity type as ${\\mathbb A}$ can be homomorphically\nmapped to ${\\mathbb A}$. It is easy to come up with examples of ${\\mathbb A}$ such that\n$\\operatorname{CSP}({\\mathbb A})$ is NP-complete and this is in a sense typical behavior\nas proved in \\cite{nesetril-random}: If $R(n,k)$ is a $k$-ary random relation on the\nset $[n]$ (with $p=1\/2$) that does not contain any elements of the form $(a,a,\\dots,a)$ for\n$a\\in A$ then\n\\begin{align}\n\\forall k\\geq 2,\\,\\lim_{n\\to\\infty}\\operatorname{Prob}(\\operatorname{CSP}([n],R(n,k))\\text{ is\nNP-complete})=1,\\label{nesetril1}\\\\\n\\forall n\\geq 2, \\lim_{k\\to\\infty}\\operatorname{Prob}(\\operatorname{CSP}([n],R(n,k))\\text{ is\nNP-complete})=1.\\label{nesetril2}\n\\end{align}\n\nThere is a reason why the authors of \\cite{nesetril-random} disallow loops: If\n${\\mathbb A}$ has only one relation $R$ and $R$ contains a loop $(a,a,\\dots,a)$ then\nevery ${\\mathbb B}$ of the same similarity type as ${\\mathbb A}$ can be homomorphically mapped to\n${\\mathbb A}$ simply by sending everything to $a$, so $\\operatorname{CSP}(A)$ is very simple to solve.\n\nGiven a target structure ${\\mathbb A}$, the \\emph{Homomorphism Extension Problem} for\n${\\mathbb A}$, denoted by $\\operatorname{EXT}({\\mathbb A})$, consists of deciding whether a given input structure\n${\\mathbb B}$ and a given partial mapping $f:{\\mathbb B}\\to {\\mathbb A}$ can be extended to a\nhomomorphism from ${\\mathbb B}$ to ${\\mathbb A}$.\n\nLet ${\\mathbb A}$ be a set and $a\\in A$. The \\emph{constant relation} $c_a$ is the unary\nrelation consisting only of $a$, i.e. $c_a=\\{(a)\\}$. When searching for a\nhomomorphism to ${\\mathbb A}$, the relation $c_a$ prescribes a set of elements of $B$\nthat must be mapped to $a$. A little thought gives us that if ${\\mathbb A}$ contains \nconstant relations for each of its elements (as is usual in the algebraic\ntreatment of CSP) then $\\operatorname{CSP}({\\mathbb A})$ and $\\operatorname{EXT}({\\mathbb A})$ are\nessentially the same problem.\n\nSince the homomorphism extension problem is quite important to algebraists, it\nmakes sense to ask what is the typical complexity of $\\operatorname{EXT}({\\mathbb A})$. We will use\nthe phrase ``$\\operatorname{EXT}({\\mathbb A})$ is almost surely NP-compete for $n$ large'' as an\nabbreviation\nfor ``For each $n\\in{\\mathbb N}$, there exists a random relational structure ${\\mathbb A}_n$ (whose precise definition\nis obvious from the context) such that we have \n\\[\n\\lim_{n\\to\\infty}\\operatorname{Prob}(\\operatorname{EXT}({\\mathbb A}_n)\\text{ is NP-complete})=1.\\text{''} \n\\]\n\nBecause additional relations do not make $\\operatorname{CSP}$ easier to solve,\nthe limit (\\ref{nesetril1}) gives us that\nthat $\\operatorname{EXT}({\\mathbb A})$ is almost surely NP-complete if ${\\mathbb A}$ is a\nlarge random relational structure with no loops and at least one relation of arity greater than\none. In the remainder of the paper we show that we can allow loops without\nmaking the problem any easier.\n\n\\section{The problem $\\operatorname{EXT}$ for random digraphs}\nWe will begin by investigating random digraphs and then generalize our findings\nto all relational structures.\n\n\\begin{theorem}\\label{thmNPgraph}\nLet $G$ be a random digraph on $n$ vertices. Then $\\operatorname{EXT}(G)$ is almost surely\nNP-complete for $n$ large.\n\\end{theorem}\n\n\\begin{proof}\n\nLet $G=(V,E)$ be a digraph. Understand $G$ as a relational structure\nand add to $G$ every constant relation possible. Let $v_1,\\dots,v_l\\in V(G)$. Consider the set \n\\[\nF_{v_1,\\dots,v_l}=\\{u\\in V(G): \\forall i, (v_i,u)\\in E(G)\\}\n\\]\nWe will call this set a \\emph{subalgebra} of $G$. \n\nFor an interested reader, we note that sets $F_{v_1,\\dots,v_l}$ are \nsubalgebras in the universal algebraic sense and our technique can be greatly\ngeneralized to all primitive positive definitions (see \\cite{BJK}). For\nour proof, however, we need a lot less: Assume that for some choice of\n$v_1,\\dots, v_l$ the subalgebra $F_{v_1,\\dots,v_l}$ induces a loopless triangle\nin $G$.\nWe claim that we can then reduce graph 3-colorability\nto $\\operatorname{EXT}(G)$, making $\\operatorname{EXT}(G)$ NP-complete. \n\n\nLet $H$ be a graph whose 3-colorability we wish to test. We then understand $H$\nas a symmetric digraph and add to $H$ new vertices $w_1,\\dots,w_l$ and new\nedges $(w_i,u)$ for each $i\\in \\{1,\\dots,n\\}$ and all $u\\in V(H)$, obtaining\nthe digraph $H'$. \nOur $\\operatorname{EXT}(G)$ instance will then\nconsist of the digraph $H'$ along with the partial map $f$ which maps each\n$w_i$ to $v_i$. Now $f$ can be extended to a homomorphism if and only if $H$\ncan be homomorphically mapped into the triangle induced by $F_{v_1,\\dots,v_l}$\nwhich happens if and only if $H$ is 3-colorable.\n\n\n\nAll we need to do now is to show that $G$ almost surely contains a\nsubalgebra that induces a triangle. Our aim, roughly speaking, is to show that $G$ almost surely\ncontains many three element subalgebras because then there is a large chance\nthat at least one of these subalgebras will be a triangle.\n\nWe will partition $V(G)$ into two\nsets $A=\\{1,\\dots,\\lfloor n\/2\\rfloor\\}$ and $B=\\{\\lceil n\/2\n\\rceil,\\dots,n\\}$. \nWe will now use points of $A$ to define subalgebras lying in $B$.\nDenote by $S_k$ the event ``$G$ contains at least $k$ disjoint three-element\nsubalgebras of the form $F_{v_1,\\dots,v_l}\\subset B$ for some $v_1,\\dots,v_l\\in\nA$.''  We\ncan write \n$$S_k=\\bigcup_{\\substack{C_1,\\dots,C_k\\subset B\\\\\n\\forall i\\neq j,\\, C_i\\cap C_j=\\emptyset\\\\\n\\forall i,\\, |C_i|=3}} S_{C_1,\\dots,C_k},\n$$\nwhere $S_{C_1,\\dots,C_k}$ is the event ``The sets\n$C_1,\\dots,C_k$ are subalgebras of $G$''. Finally,\ndenote by $T_{C_1,\\dots,C_k}$ the event ``There exists an $i\\in\\{1,2.\\dots,k\\}$\nsuch that the set $C_i$ induces a triangle subgraph of $G$.''\n\nSince a probability that a fixed $C_i$ induces a triangle is $p^6(1-p^3)$, \nthe probability of the event $T_{C_1,\\dots,C_k}$ is (for $C_1,\\dots,C_k$\npairwise disjoint three element sets) \n$$\n\\operatorname{Prob}(T_{C_1,\\dots,C_k})=1-(1-p^6(1-p^3))^k,\n$$\nwhich tends to 1 when $k$ goes to infinity.\n\nObserve that the event $S_{C_1,\\dots,C_k}$ is independent from the event\n$T_{C_1,\\dots,C_k}$ for each choice of $C_1,\\dots,C_k\\subset B$ \nsince both events talk about disjoint sets of edges of $G$. \n\nAssume for a moment that for all $k\\in {\\mathbb N}$ the value of $\\operatorname{Prob}(S_k)$ tends to 1\nas $n$ tends to infinity. Then, given an $\\epsilon>0$, we choose $k$ so that\n$\\operatorname{Prob}(T_{C_1,\\dots,C_k})\\geq 1-\\epsilon$. When $n$ is large enough, \nthe digraph $G$ contains some $k$ pairwise disjoint\nthree element subalgebras $C_1,\\dots,C_k$\nwith probability at least $1-\\epsilon$. The probability that one of the sets\n$C_1,\\dots,C_k$ them induces a triangle is $T_{C_1,\\dots,C_k}\\geq\n1-\\epsilon$. Thus we get an NP-complete CSP problem with probability at least $(1-\\epsilon)^2>\n1-2\\epsilon$ and since $\\epsilon$ was arbitrary, we see that for large $n$ the\nhomomorphism extension problem is almost surely NP-complete.\n\nIt remains to show $\\lim_{n\\to\\infty}\\operatorname{Prob}(S_k)=1$ for all $k$. Fix the value\nof $k$. For each value of $n$, let $l$ be the integer satisfying $n p^l\\geq 1>n p^{l+1}$. We will now search\nfor the three element subalgebras of $B$ for $n$ large. We proceed in steps: Assume that after $i$\nsteps we have already found $m$ such subalgebras $C_1,\\dots,C_m$. In the $(i+1)$-th step, \nwe take the vertices $1+il,2+il,\\dots,l+il$\nof $A$ and consider the subalgebra $F_{1+il,2+il,\\dots,l+il}$. If this\nsubalgebra lies in $B$, has size three and is disjoint with all the sets $C_1,\\dots,C_m$, we\nlet $C_{m+1}=F_{1+il,2+il,\\dots,l+il}$, increase $m$ by one and continue with\nthe next step. Otherwise, $F_{1+il,2+il,\\dots,l+il}$ is not a good candidate for\n$C_{m+1}$, so we leave $m$ unchanged and continue with the next step.\n\nWhat is the probability that we find the $(m+1)$-th\nsubalgebra in a given step? Every vertex of $G$ is in $F_{1+il,2+il,\\dots,l+il}$ with the\nprobability $p^l$. The probability that $F_{1+il,2+il,\\dots,l+il}$ consists of\nthree yet-unused vertices of $B$ is then equal to\n\\[\nq={|B|-3\\cdot m \\choose 3}p^{3l}(1-p^l)^{n-3}\\geq\n\\frac{(n\/2-3m-3)^3}{6}p^{3l}(1-p^l)^n\n\\]\nIf $m\\geq k$, we have already won, so assume $m<k$:\n\\[\nq\\geq \\frac{(n\/2-3k)^3}{6}p^{3l}(1-p^l)^n=\\frac{(1\/2-3k\/n)^3}{6}\nn^3p^{3l}(1-p^l)^n\n\\]\nNow let $r=\\frac{(1\/2-3k\/n)^3}6$ and observe that $r>0$ for $n$ large\nenough. Using the the inequalities $n p^l\\geq 1>n p^{l+1}$ we have: \n\\[\nq \\geq r n^3 p^{3l}(1-p^l)^n \\geq r (1-p^l)^n> r \\left(1-\\frac{1}{pn}\\right)^n.\n\\]\nThe lower\nbound on $q$ tends to $r\/e^{1\/p}$ as $n$ tends to infinity, so there exists a $\\delta$ such that $q>\\delta>0$\nfor all $n$ large enough.\n\nTherefore, the probability of producing a new three-element subalgebra in a\ngiven step is at\nleast $\\delta>0$ and this bound does not depend on the number of subalgebras\nwe have already found. Now observe that $l$\nis approximately $\\log_{1\/p} n$ and therefore we have enough vertices in $A$ for\napproximately $s=\\frac{n}{2\\log_{1\/p}n}$ steps. If we choose $n$ large enough, we can have $s$ as\nlarge as we want and so the probability of finding at least $k$ subalgebras\ncan be arbitrarily close to 1. Therefore, $\\lim_{n\\to\\infty} \\operatorname{Prob}(S_k)=1$, concluding the\nproof.\n\\end{proof}\n\n\\section{Random relational structures}\nIt is easy to see that if ${\\mathbb A}$ is a relational structure with unary\nrelations only then $\\operatorname{EXT}({\\mathbb A})$ is always polynomial. We would now like to\ninvestigate the case of relations of arity greater than two. Intuition tells us\nthat greater arity means greater complexity. The intuition is right.\n\n\\begin{lemma}\nLet $l\\geq 2$, $n$ be large and let ${\\mathbb A}=([n],S)$ be a relational structure with\n$S$ a random $l$-ary relation. Then\nthe homomorphism extension problem $\\operatorname{CSP}({\\mathbb A})$ is almost surely NP-complete.\n\\end{lemma}\n\\begin{proof}\nWe have proven the result for $l=2$. If $l>2$, consider the binary relational structure ${\\mathbb B}=([n],R)$ where \n$R=\\{(x,y)\\in [n]^2: (x,y,1,1,\\dots,1)\\in S\\}$. It is easy to see that if $S$ is\na random $l$-ary relation then ${\\mathbb B}$ is a random \ndigraph where each edge exists with the probability $p$. From\nTheorem~\\ref{thmNPgraph} we see that\n$\\operatorname{EXT}({\\mathbb B})$ is almost surely NP-complete. We will now show how to reduce\n$\\operatorname{EXT}({\\mathbb B})$ to $\\operatorname{EXT}({\\mathbb A})$ in polynomial time, proving that $\\operatorname{EXT}({\\mathbb A})$\nis almost surely NP-complete.\n\nUsing algebraic tools, the reduction of $\\operatorname{EXT}({\\mathbb B})$ to $\\operatorname{EXT}({\\mathbb A})$ follows from the fact that\n$R$ is defined by a primitive positive formula that uses only $S$ and the constant $1$.\nHowever, we will provide an elementary reduction here: Let ${\\mathbb C}=(C,T)$ be a\nrelational structure with a single binary relation $T$ and let $f:C\\to [n]$ be a \npartial mapping. We add to $C$ a new element $e$, construct the\n$l$-ary relation $U=\\{(x,y,e,e,\\dots,e):(x,y)\\in T\\}$ and the partial mapping\n$g:C\\cup \\{e\\}\\to [n]$ so that $g_{|C}=f$ and $g(e)=1$. A little thought gives\nus that $g$ can be extended to a homomorphism $(C\\cup\\{e\\},U)\\to {\\mathbb A}$ if and\nonly if $f$ can be extended to a homomorphism $(C,T)\\to {\\mathbb B}$, concluding the\nproof.\n\\end{proof}\n\nAdditional relations in ${\\mathbb A}$ do not make $\\operatorname{EXT}({\\mathbb A})$ easier, so we have the most\ngeneral version of our NP-completeness result:\n\n\\begin{corollary}\nLet ${\\mathbb A}$ be the relational structure $([n],\\{R_i:i\\in I\\})$ where at least one\n$R_i$ is a random\nrelation of arity greater than one. Then $\\operatorname{EXT}({\\mathbb A})$ is \nalmost surely NP-complete for $n$ large.\n\\end{corollary}\n\nAs a final note, we will now prove the analogue of the limit~(\\ref{nesetril2})\nfor $\\operatorname{EXT}$.\n\n\\begin{corollary}\nLet us fix a set $A$ of at least two elements and let ${\\mathbb A}=(A,R)$ be \na relational structure with $R$\nrandom $k$-ary relation. Then $\\operatorname{EXT}({\\mathbb A})$ is almost surely NP-complete for $k$\nlarge.\n\\end{corollary}\n\\begin{proof}\nAssume first that $k$ is even and let $m=k\/2$.  \n\nConsider the relational structure ${\\mathbb B}=(A^m,S)$ with \n$$\nS=\\{((a_1,\\dots,a_m),(a_{m+1},\\dots,a_{2m})):(a_1,\\dots,a_{2m})\\in R\\}.\n$$\nIt is straightforward to prove that $S$ is a binary random relation on $A^m$ and therefore $\\operatorname{EXT}({\\mathbb B})$\nis almost surely NP-complete for large even $k$. What is more, $\\operatorname{EXT}({\\mathbb B})$ can\nbe easily reduced to $\\operatorname{EXT}({\\mathbb A})$: If ${\\mathbb C}=(C,T)$ is a relational\nstructure with $T$ binary and $f:C\\to A^m$ is a partial mapping, \nwe construct the structure ${\\mathbb C}'=(C',T')$ with \n\\begin{align*}\nC'&=\\{(c,i):c\\in C, i\\in \\{1,\\dots,m\\}\\},\\\\\nT'&=\\{((c,1),\\dots,(c,m),(d,1),\\dots,(d,m)):(c,d)\\in T\\}\n\\end{align*}\nand a partial mapping $g:C'\\to A$ such that $g(c,i)=a_i$ whenever $f(c)$ is defined and equal to\n$(a_1,\\dots,a_m)$.\n\nIt is easy to see that $g$ can be extended to a homomorphism from ${\\mathbb C}'$ to ${\\mathbb A}$\nif and only if $f$ can be extended to a homomorphism from ${\\mathbb C}$ to ${\\mathbb A}$. \n\nIn the case that $k=2m+1$, we fix an $e\\in A$, choose\n${\\mathbb B}=(A^m,S)$ with\n$$\nS=\\{((a_1,\\dots,a_m),(a_{m+1},\\dots,a_{2m}):(a_{1},\\dots,a_{2m},e)\\in R\\}\n$$\nand proceed similarly to the previous case. \n\nWe see that for a large enough $k$, no matter if it is odd or even, the problem\n$\\operatorname{EXT}({\\mathbb B})$ is almost surely NP-complete.\n\\end{proof}\n\n\\section{Conclusions}\nWe have shown that the homomorphism extension problem is almost surely\nNP-complete for large relational structures (assuming we have at least one\nnon-unary relation). In a sense, our result is not surprising since the relational\nstructures we consider are very dense, so it stands to a reason that we can \nfind hard instances most of the time.\n\nIt might therefore be interesting to see what is the complexity of CSP or EXT\nfor large structures obtained by other random processes, particularly when\nrelations are sparse. Such structures might better correspond to ``typical''\ncases of CSP or EXT encountered in practice. Some such results already exist;\nsee \\cite{nesetril-luczak-projective} for a criterion on the random graph\nprocess to almost surely produce projective graphs (if $G$ is\nprojective then $\\operatorname{EXT}(G)$ is NP-complete, see \\cite{BJK}). Our guess is that\nboth CSP and EXT will remain to be almost surely NP-complete in all the\nnontrivial cases.\n\n\n\\section{Acknowledgments}\nThe research was supported by the GA\\v CR project GA\\v CR 201\/09\/H012, by the Charles\nUniversity project GA UK 67410 and by the grant SVV-2011-263317. The \nauthor would like to thank Libor Barto for suggesting this problem.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the very beginning of the creation of quantum mechanics, the question of its formulation\nin terms of the distribution function on the phase space, like the classical kinetic theory, \nhas attracted the attention of many scientists, despite the fact that the Heisenberg uncertainty relation \nprohibits the existence  of the joint distribution function of the position and momentum in the quantum case.\n\nA great success in this connection was the introduction of the Wigner function \\cite{Wigner32}\nand the writing of the dynamic equation for it \\cite{Wigner32,Moyal1949}.\nFor the $N-$dimensional system the Wigner function is introduced as the Weyl symbol \\cite{Weyl1927} \nof the  density matrix $\\hat\\rho(t)$ of the state\n\\be\t\t\t\\label{WigDef}\nW(\\mathbf{q},\\mathbf{P},t)=\\frac{1}{(2\\pi\\hbar)^N}\n\\int\\left<\\left.\\mathbf{q}-\\frac{\\mathbf u}{2}\\right|\\hat\\rho(t)\\left|\\mathbf{q}+\\frac{\\mathbf u}{2}\\right.\\right>\n\\exp\\left(\\frac{\\rmi}{\\hbar}\\mathbf{u}\\mathbf{P}\\right)\\rmd ^Nu,\n\\ee\nwhere $\\mbf P$ is the generalized momentum \ncorresponding to the generalized momentum operator $\\hat{\\mbf P}=-\\rmi\\hbar\\partial\/\\partial{\\mbf q}$.\nThe transformation inverse to (\\ref{WigDef}) has the form:\n\\be\t\t\t\\label{RhoFromW}\n\\left<\\mbf q|\\hat\\rho(t)|\\mbf q'\\right>=\\int W\\left(\\frac{\\mbf q+\\mbf q'}{2},\\mbf P,t\\right)\n\\exp\\left(\\rmi\\mbf P\\frac{\\mbf q-\\mbf q'}{\\hbar}\\right)d^NP.\n\\ee\nDespite the fact that the Wigner function can take negative values, it is successfully applied\nin many applications since the 1950s (see, e.g., \\cite{Silin1, Silin2, Silin3, Silin4, Landau}) and to the present.\nThe properties of the Wigner function were considered, e.g., in \\cite{OConnellWignerPhysRep}.\n\nAt the same time, if we smoothly average the Wigner function on the scales of the hyper-volume\n$\\hbar^N$ of phase space, then it can be made nonnegative.\nThis way in the article \\cite{Husimi40} and later in \\cite{Kano1965} the Husimi function \n(the so called $Q-$function) was introduced  \n\\be\t\t\t\\label{QDef}\nQ(\\mathbf{q},\\mathbf{P},t)=\\frac{1}{(2\\pi\\hbar)^N}\\left<\\bm\\alpha|\\hat\\rho(t)|\\bm\\alpha\\right>,\n\\ee\nrepresenting up to the normalization factor the transition probability of a quantum system \nfrom the state $\\hat\\rho(t)$ into a coherent state\n$|\\bm\\alpha\\rangle$ \\cite{GlauberPhysRevLett1963,GlauberPhysRev1963}, where \n$\\bm\\alpha=(2\\hbar)^{-1\/2}(\\lambda^{1\/2}\\mbf q+\\rmi\\lambda^{-1\/2}\\mbf P)$, \n$\\lambda=m\\omega$. Up to the inessential phase factor the state $|\\bm\\alpha\\rangle$\nis defined in the position representation as follows:\n\\be\t\t\t\\label{CohStateDef}\n\\langle\\mbf q'|\\bm\\alpha\\rangle=\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{N\/4}\n\\exp\\left[-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q')^2-\\frac{\\rmi}{\\hbar}\\mbf P(\\mbf q-\\mbf q')\\right].\n\\ee\nAs is known, the state $|\\bm\\alpha\\rangle$ minimizes the Heisenberg uncertainty relation, \nand the remarkable property of the $Q-$function is its connection with the Wigner function \nby means of the formula: \n\\be\t\t\t\\label{WignerToHusimi}\nQ(\\mathbf{q},\\mathbf{P},t)=\\frac{1}{(\\pi\\hbar)^N}\\int\\exp\\left(-\\frac{\\lambda}{\\hbar}(\\mbf q-\\mbf q')^2\n-\\frac{1}{\\lambda\\hbar}(\\mbf P-\\mbf P')^2\\right)W(\\mbf q',\\mbf P',t)d^Nq'd^NP',\n\\ee\nrepresenting the averaging of the function $W(\\mbf q,\\mbf P,t)$ \nin the phase space with respect to a Gaussian distribution centered at the point $(\\mbf q,\\mbf P)$.\nFormula (\\ref{WignerToHusimi}) can be rewritten in the operator form,\n\\be\t\t\t\\label{WigToHusOperator}\nQ(\\mathbf{q},\\mathbf{P},t)=\\exp\\left(\\frac{\\hbar}{4\\lambda}\\partial_{\\mbf q}^2\n+\\frac{\\lambda\\hbar}{4}\\partial_{\\mbf P}^2\\right)W(\\mbf q,\\mbf P,t),\n\\ee\nand the inverse transform of (\\ref{WigToHusOperator}) is obviously has the shape:\n\\be\t\t\t\\label{HusToWigOperator}\nW(\\mathbf{q},\\mathbf{P},t)=\\exp\\left(-\\frac{\\hbar}{4\\lambda}\\partial_{\\mbf q}^2\n-\\frac{\\lambda\\hbar}{4}\\partial_{\\mbf P}^2\\right)Q(\\mbf q,\\mbf P,t).\n\\ee\nAlso the inverse transform of (\\ref{WignerToHusimi}) can be expressed\nas a repeated integral,\n\\be\t\t\t\\label{HusimiToWigner}\nW(\\mathbf{q},\\mathbf{P},t)=\\int\\frac{d^Nud^Nv}{(2\\pi)^{2N}}\\int\\exp\\left[\\frac{\\hbar\\mbf u^2}{4\\lambda}\n+\\rmi\\mbf u(\\mbf q-\\mbf q')+\\frac{\\lambda\\hbar\\mbf v^2}{4} +\\rmi\\mbf v(\\mbf P-\\mbf P')\\right]Q(\\mbf q',\\mbf P',t)d^Nq'd^NP'.\n\\ee \n\nExpansion of  formula (\\ref{WignerToHusimi}) to the classical (non-quantum) case\nallows to determine the Husimi function $Q_{\\mathrm{cl}}(\\mbf q,\\mbf P,t)$ of the state \nof the classical system having described by the classical distribution function\n$W_{\\mathrm{cl}}(\\mbf q,\\mbf P,t)$ as the overlap with a Gaussian distribution.\n\nConsider the motion of a quantum particle having a spin in the \nelectromagnetic field with the vector potential $\\mathbf{A}(\\mathbf{q},t)$\nand the scalar potential $\\varphi(\\mathbf{q},t)$. \nAs it is known, the Hamiltonian of such a system has the form \\cite{LandauIII}\n\\be\t\t\t\\label{Hamiltonian}\n\\hat H=\\frac{1}{2m}\\left(\\hat{\\mathbf P}-\\frac{e}{c}\\mathbf{A}\\right)^2\n+e\\varphi- \\hat{\\bm\\kappa}  \\mathbf{B},\n\\ee\nwhere $\\hat{\\mathbf P}=-i\\hbar\\partial\/\\partial\\mathbf{q}$ is a generalized momentum operator,\n$m$ and $e$ are mass and charge of the particle, $\\mathbf{B}=\\mathrm{rot}\\mathbf{A}$ \nis a magnetic field strength, $\\hat{\\bm\\kappa}$ is an operator \nof quantum-mechanical  magnetic moment\n\\be\t\t\t\\label{Moment}\n\\hat{\\bm\\kappa}=\\frac{\\kappa}{s}\\hat{\\mathbf{s}},\n\\ee\nwhere $s$ is a spin of the particle, $\\hat{\\mathbf{s}}$ is a spin operator,\nand $\\kappa$ is the value of the intrinsic magnetic moment of the particle.\n\nFrom the classical electrodynamics it is known that \npotentials of the field are defined only up to the gauge transformation\n\\cite{LandauII}\n\\be\t\t\t\\label{eq3}\n\\mathbf{A} ~~\\rightarrow ~~\\mathbf{A} +\\nabla \\chi,\n~~~~\n\\varphi~~\\rightarrow ~~\\varphi-\\frac{1}{c}\\frac{\\partial\\chi}{\\partial t},\n\\ee\nwhere $\\chi$ is an arbitrary function of spatial coordinates and time.\n\nSince the electric field intensity $\\mathbf{E}$ and the magnetic field strength\n$\\mathbf{B}$ are defined in terms of the potentials as:\n\\be\t\t\t\\label{eq4}\n\\mathbf{E}=-\\mathrm{grad}\\varphi-\\frac{1}{c}\\frac{\\partial}{\\partial t}\\mathbf{A},\n~~~~\n\\mathbf{B}=\\mathrm{rot}\\mathbf{A},\n\\ee\nthen the gauge transformation (\\ref{eq3}) does not affect the values \nof $\\mbf E$ and $\\mbf B$.\nTherefore the part of Hamiltonian (\\ref{Hamiltonian}) responsible for\nthe interaction of the spin with the magnetic field is independent \non the gauge transformation, and  we can restrict our considerations only to the case\nof $s=0$. The generalization to the non-zero spin particles is straightforward \n(see \\cite{Korarticle9,Korarticle14,Korarticle12}).\n\n\n\nThe requirement of invariance of the Schr\\\"odinger equation \nunder the gauge transformation simultaneously with the gauge-independence\nof ``probability density''\\, $|\\Psi|^2$ leads us to the form of the conversion\nof the wave function \\cite{LandauIII}:\n\\be\t\t\t\\label{eq5}\n\\Psi ~~\\rightarrow ~~\\exp\\left(\\frac{\\rmi e}{c\\hbar}\\chi \\right)\\Psi.\n\\ee\nAccordingly, the conversions of the density matrix of the state of the \nsystem under the gauge transformation acquires the form:\n\\be\t\t\t\\label{eq6}\n\\hat\\rho_{\\mathrm c}=\n\\exp\\left(\\frac{\\rmi e}{c\\hbar}\\chi \\right)\\hat\\rho\\,\\exp\\left(-\\frac{\\rmi e}{c\\hbar}\\chi \\right).\n\\ee\n\nIn \\cite{Stratonovich2} the gauge-independent Wigner function was constructed, and its \nevolution equation was derived in \\cite{Serimaa1986}. The gauge-invariance in the \ntomographic probability representation of quantum mechanics was considered in \\cite{Korarticle12}\n(see review articles \\cite{IbortPhysScr,MankoMankoFoundPhys2009} about the \ntomographic probability representation).\n\nThe aim of this work is introduction of\ngauge-independent Husimi function ($Q-$function) of states of charged quantum \nparticles in the electro-magnetic field, and  is derivation of the evolution equation for such function.\n\n\n\\section{\\label{Art08Section2}Gauge-independent Husimi function}\nFor the construction of quantum Husimi representations, \nin which the evolution equation would be gauge-independent, \nwe need to introduce gauge-independent quantum Husimi functions.\nThis can be done with the help of a gauge-independent Wigner function\nobtained in \\cite{Stratonovich2},\n\\be\nW_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t) \n=\\frac{1}{(2\\pi\\hbar)^{3}}\\int\n\\exp\\left(\\frac{\\rmi }{\\hbar}\\mathbf{u}\\left\\{\\mathbf{p}\n+\\frac{e}{c}\\int_{-1\/2}^{1\/2} \\rmd\\tau \\mathbf{A}(\\mathbf{q}+\\tau\\mathbf{u},\\,t)\\right\\}\\right)\n\\rho\\left(\\mathbf{q}-\\frac{\\mathbf u}{2},\\,\\mathbf{q}+\\frac{\\mathbf u}{2},\\,t\\right)\n\\rmd^3u,\n\t\t\t\\label{WigNew}\n\\ee\nwhere $\\mathbf{p}$ is a kinetic  momentum.\n\nThe gauge-independent Husimi function $ Q_\\mathrm{g}(\\mbf q,\\mbf p,t)$ should be introduced\nusing a formula similar to (\\ref{WignerToHusimi})\n\\be\t\t\t\\label{WignerGToHusimiG}\nQ_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=\\frac{1}{(\\pi\\hbar)^N}\\int\\exp\\left(-\\frac{\\lambda}{\\hbar}(\\mbf q-\\mbf q')^2\n-\\frac{1}{\\lambda\\hbar}(\\mbf p-\\mbf p')^2\\right)W_\\mathrm{g}(\\mbf q',\\mbf p',t)d^Nq'd^Np'.\n\\ee\nWith this definition the formulas (\\ref{WigToHusOperator},\\,\\ref{HusToWigOperator},\\,\\ref{HusimiToWigner})\nof the relations between $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ and $W_\\mathrm{g}(\\mbf q',\\mbf p',t)$ \nremain valid if we replace the generalized momentum $\\mbf P$ by the kinetic momentum $\\mbf p$ in them.\n\nCombining formulas (\\ref{WigNew}) and (\\ref{WignerGToHusimiG}) we can write\n\\be\t\t\t\\label{QgFromRho}\nQ_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=\\int \\langle\\mbf q_1|\\hat\\rho(t)|\\mbf q_2\\rangle \n\\langle\\mbf q_2|\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle d^3q_1d^3q_2,\n\\ee\nwhere we introduce the matrix element for the corresponding dequantizer operator\n\\bea\n\\langle\\mbf q_2|\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle&=&\n\\frac{1}{(2\\pi\\hbar)^3}\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{3\/2}\n\\exp\\Bigg\\{-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_2)^2-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_1)^2\n\\nonumber \\\\[3mm]\n&+&\\frac{\\rmi}{\\hbar}(\\mbf q_2-\\mbf q_1)\\left[\\mbf p\n+\\frac{e}{c}\\int_{-1\/2}^{1\/2}\\mbf A\\left(\\frac{\\mbf q_2+\\mbf q_1}{2}+\\tau(\\mbf q_2-\\mbf q_1),t\\right)d\\tau\\right]\\Bigg\\}.\n\t\t\t\\label{matrUQg}\n\\eea\nFrom (\\ref{matrUQg}) we can see that $\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ is Hermitian  operator, \nso the Husimi function $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ is real.\n\nThe explicit form of the operator $\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ can be written in the integral form\n\\be\t\t\t\\label{UQgint}\n\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)=\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^6}\n\\exp\\left[-\\frac{\\lambda\\mbf u^2}{4\\hbar}-\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf u\\mbf p+\\mbf v\\mbf q)\\right]\n\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\left(\\hat{\\mbf P}-\\frac{e}{c}\\mbf A(\\hat{\\mbf q},t)\\right)\n+\\mbf v\\hat{\\mbf q}\\right]\\right\\}.\n\\ee \n\nThe transformation inverse to (\\ref{QgFromRho}) can be expressed using the matrix element of \nquantizer operator $\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\n\\be\t\t\t\\label{RhoFromQg}\n\\langle\\mbf q_1|\\hat\\rho(t)|\\mbf q_2\\rangle\n=\\int \\langle\\mbf q_1|\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_2\\rangle\nQ_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t) d^3qd^3p,\n\\ee\nwhere\n\\bea\n\\langle\\mbf q_1|\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_2\\rangle&=&\n\\int\\frac{d^3v}{(2\\pi\\hbar)^3}\\exp\\Bigg[\\frac{\\lambda(\\mbf q_2-\\mbf q_1)^2}{4\\hbar}\n+\\frac{\\mbf v^2}{4\\hbar\\lambda}-\\frac{\\rmi\\mbf v}{2\\hbar}(2\\mbf q-\\mbf q_1-\\mbf q_2)\n-\\frac{\\rmi\\mbf p}{\\hbar}(\\mbf q_2-\\mbf q_1) \\nonumber \\\\[3mm]\n&-&\\frac{\\rmi e}{\\hbar c}(\\mbf q_2-\\mbf q_1)\\int_{-1\/2}^{1\/2}d\\tau\n\\mbf A\\left(\\frac{\\mbf q_2+\\mbf q_1}{2}+\\tau(\\mbf q_2-\\mbf q_1),t\\right)\n\\Bigg].\n\t\t\t\\label{matrDg}\n\\eea\nIn these formulas it is assumed that $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)\\in S(\\mathbb{R}^6)$,\nwhere $S(\\mathbb{R}^6)$ is a Schwartz space, and first we take the integral over $d^3q$,\nand after that the integrals over $d^3p$ and $d^3v$ are taken.\n\nAssuming a special order of integration, the explicit form of the operator \n$\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ can be written as:\n\\be\t\t\t\\label{DQgint}\n\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)=\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^3}\n\\exp\\left[\\frac{\\lambda \\mbf u^2}{4\\hbar}+\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf{up}+\\mbf{vq})\\right]\n\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\left(\\hat{\\mbf P}-\\frac{e}{c}\\mbf A(\\hat{\\mbf q},t)\\right)\n+\\mbf v\\hat{\\mbf q}\\right]\\right\\}.\n\\ee\n\n\n\n\\section{\\label{Art08Section3}Evolution equation for the gauge-independent Husimi function}\nTo begin with, let us recall the Liouville equation in the electro-magnetic field \nfor the classical distribution function.\nFor the classical ensemble of non-interacting particles with \nmass $m$ and charge $e$ this equation in the phase space has the form:\n\\be\n\\left\\{\\partial_ t+\n\\frac{\\mathbf{p}}{m}\\partial_{\\mathbf{q}}\n+e\\left(\\mathbf{E}(\\mathbf{q},t)\n+\\frac{1}{mc}[\\mathbf{p}\\times\\mathbf{B}(\\mathbf{q},t)]\\right)\n\\partial_{\\mathbf{p}}\\right\\}W_\\mathrm{cl}(\\mathbf{q},\\mathbf{p},t)=0,\n\t\t\t\\label{Liouville}\n\\ee\nwhere $\\mathbf{p}$ is a kinetic momentum, $\\mathbf{E}(\\mathbf{q},t)$ and \n$\\mathbf{B}(\\mathbf{q},t)$ are electric and magnetic fields, \ndefined by formulas (\\ref{eq4}), $W_\\mathrm{cl}(\\mathbf{q},\\mathbf{p},t)$\nis a distribution function of non-interacting particles.\n\nThe distribution function $W_\\mathrm{cl}(\\mathbf{q},\\mathbf{p},t)$ is independent on \nthe gauge transformation \\cite{LandauII} \nbecause the Liouville equation (\\ref{Liouville}) includes \nonly gauge-independent intensities of the electro-magnetic field.\n\n\nGauge-independent Moyal equation for the Wigner function $W_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ has the form \\cite{Serimaa1986}:\n\\be\t\t\t\\label{EqWigNew}\n\\left\\{\\partial_t+\\frac{1}{m}\\left(\\mathbf{p}+\\triangle\\tilde{\\mathbf p}\\right)\\partial_{\\mathbf{q}}\n+ e\\left(\\tilde{\\mathbf{E}} +\\frac{1}{mc}\n\\left[\\left(\\mathbf{p}+\\triangle\\tilde{\\mathbf p}\\right)\\times\\tilde{\\mathbf{B}} \\right]\\right)\n\\partial_{\\mathbf p}\n\\right\\}W_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=0,\n\\ee\nwhere\n\\bdm\n\\triangle\\tilde{\\mathbf p}=-\\frac{e}{c}\\frac{\\hbar}{\\rmi }\n\\left[\n\\frac{\\partial}{\\partial\\mathbf{p}} \\times \\int_{-1\/2}^{1\/2}\n\\rmd\\tau\\,\\tau \n\\mathbf{B}\n\\left(\n\\mathbf{q}+\\rmi\\hbar\\tau\\frac{\\partial}{\\partial\\mathbf{p}},\\,t\n\\right)\n\\right],\n\\edm\n\\bdm\n\\tilde{\\mathbf E}=\\int_{-1\/2}^{1\/2}\n\\rmd\\tau\\,\n\\mathbf{E}\n\\left(\n\\mathbf{q}+\\rmi \\hbar\\tau\\frac{\\partial}{\\partial\\mathbf{p}},\\,t\n\\right),\n~~~~\n\\tilde{\\mathbf B}=\\int_{-1\/2}^{1\/2}\n\\rmd\\tau\\,\n\\mathbf{B}\n\\left(\n\\mathbf{q}+\\rmi \\hbar\\tau\\frac{\\partial}{\\partial\\mathbf{p}},\\,t\n\\right).\n\\edm\nThis equation in the classical limit $\\hbar\\to 0$ is converted into \nLiouville equation (\\ref{Liouville}).\n\nSince the relations between the functions $W_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ and\n$Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ are the same as between $W(\\mathbf{q},\\mathbf{p},t)$ and\n$Q(\\mathbf{q},\\mathbf{p},t)$, then the correspondence rules between\noperators acting on the Wigner function and the Husimi function do not change.\nConsequently, we can write:\n\\be\t\t\\label{CorrespRulesWgQg}\n\\begin{array} {lcl} \n\\mbf q\\, W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_\\mbf{q}\\right)Q(\\mathbf{q},\\mathbf{p},t),\n\\\\\n\\mbf p\\, W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\left(\\mbf p+\\frac{\\hbar\\lambda}{2}\\partial_\\mbf{p}\\right)Q(\\mathbf{q},\\mathbf{p},t),\n\\\\\n\\partial_{\\mbf{q}} W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\partial_{\\mbf{q}}Q(\\mathbf{q},\\mathbf{p},t),\n\\\\\n\\partial_{\\mbf{p}} W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\partial_{\\mbf{p}}Q(\\mathbf{q},\\mathbf{p},t).\n\\end{array}\n\\ee\nWith the help of (\\ref{CorrespRulesWgQg}) equation (\\ref{EqWigNew}) is transformed to\nthe evolution equation for the  gauge-independent Husimi function $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$\n\\bea\n&&\\bigg\\{\\partial_t+\\frac{1}{m}\\left(\\mbf p+\\frac{\\hbar\\lambda}{2}\\partial_{\\mbf p}\n+\\big[\\triangle\\tilde{\\mbf{p}}\\big]_Q\\right)\\partial_{\\mbf q} \\nonumber \\\\[3mm]\n&&~~~~~~+e\\left(\\big[\\tilde{\\mbf E}\\big]_Q\n+\\frac{1}{mc}\\left[\\left(\\mbf p+\\frac{\\hbar\\lambda}{2}\\partial_{\\mbf p}\n+\\big[\\triangle\\tilde{\\mbf{p}}\\big]_Q\\right)\\times\\big[\\tilde{\\mbf B}\\big]_Q\\right]\\right)\\partial_{\\mbf p}\n\\bigg\\}Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=0,\n\t\t\t\\label{EqforQg}\n\\eea\nwhere\n\\bea\n&&\\big[\\triangle\\tilde{\\mbf{p}}\\big]_Q=-\\frac{e}{c}\\frac{\\hbar}{\\rmi}\\left[\\partial_{\\mbf p}\\times\n\\int_{-1\/2}^{1\/2}\\mbf B\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_{\\mbf{q}}\n+\\rmi\\hbar\\tau\\partial_{\\mbf{p}},t\\right)\\tau d\\tau\\right],\n\\nonumber \\\\[3mm]\n&&\\big[\\tilde{\\mbf E}\\big]_Q=\\int_{-1\/2}^{1\/2}\\mbf E\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_{\\mbf{q}}\n+\\rmi\\hbar\\tau\\partial_{\\mbf{p}},t\\right)d\\tau,\n\\nonumber \\\\[3mm]\n&&\\big[\\tilde{\\mbf B}\\big]_Q=\\int_{-1\/2}^{1\/2}\\mbf B\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_{\\mbf{q}}\n+\\rmi\\hbar\\tau\\partial_{\\mbf{p}},t\\right)d\\tau.\n\\eea\nAs it should be, equation (\\ref{EqforQg}) in the classical limit $\\hbar\\to0$ is converted \ninto the Liouville equation (\\ref{Liouville}).\n\n\\section{\\label{Art08Section4}Non-Stratonovich type of gauge-independent \\\\\nWigner and Husimi functions}\n\nIn the previous sections we considered the Wigner and Husimi functions  on the basis of the definition \nof Stratonovich \\cite{Stratonovich2}.\nHowever, it is possible to introduce a gauge-independent Wigner function and its corresponding Husimi function\naccording to (\\ref{WignerToHusimi}, \\ref{WigToHusOperator}) by other ways.\n\nLet us define the gauge-invariant dequantizer for the new Wigner function \n$\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$ as follows:\n\\bea\n\\hat U_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)&=&\\exp\\left[\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]\n\\nonumber \\\\[3mm]\n&\\times&\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^6}\n\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\left(\\hat{\\mbf P}-\\mbf p\\right)+\\mbf v(\\hat{\\mbf q}-\\mbf q)\\right]\\right\\}\n\\exp\\left[-\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]. \n\t\t\t\\label{newDequantW}\n\\eea\nSince for the Wigner function the dequantizer-quantizer scheme is self-dual, \nthen the following equality takes place for the corresponding quantizer:\n$\\hat D_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)=\n(2\\pi\\hbar)^3\\hat U_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$.\nCalculation of the matrix element of (\\ref{newDequantW}) yields\n\\bea\n\\langle\\mbf q_2|\\hat U_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle&=&\n(2\\pi\\hbar)^{-3}\\delta\\left(\\mbf q-\\frac{\\mbf q_2+\\mbf q_1}{2}\\right) \n\\nonumber \\\\[3mm]\n&\\times&\\exp\\left[\\frac{\\rmi\\mbf p}{\\hbar}(\\mbf q_2-\\mbf q_1)\n+\\frac{\\rmi e}{\\hbar c}\\mbf q_2\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_2,t)\n-\\frac{\\rmi e}{\\hbar c}\\mbf q_1\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_1,t)\\right].\n\t\t\t\\label{matrnewDequantW}\n\\eea\nTaking into account (\\ref{WignerToHusimi}), we find the dequantizer for the corresponding\nHusimi function $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\n\\bea\n\\hat U_\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)&=&\\exp\\left[\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]\n\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^6}\n\\exp\\left[-\\frac{\\lambda\\mbf u^2}{4\\hbar}-\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf{up}+\\mbf{vq})\\right]\n\\nonumber \\\\[3mm]\n&\\times&\n\\exp\\left[\\frac{\\rmi}{\\hbar}(\\mbf u\\hat{\\mbf P}+\\mbf v\\hat{\\mbf q})\\right]\n\\exp\\left[-\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]. \n\t\t\t\\label{newDequantQ}\n\\eea\nThe matrix element (\\ref{newDequantQ}) is obviously equal to the following:\n\\bea\n\\langle\\mbf q_2|\\hat U_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle&=&\n\\frac{1}{(2\\pi\\hbar)^3}\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{3\/2}\n\\exp\\bigg[-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_2)^2-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_1)^2\n\\nonumber \\\\[3mm]\n&+&\\frac{\\rmi}{\\hbar}(\\mbf q_2-\\mbf q_1)\\mbf p\n+\\frac{\\rmi e}{\\hbar c}\\mbf q_2\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_2,t)\n-\\frac{\\rmi e}{\\hbar c}\\mbf q_1\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_1,t) \\bigg].\n\t\t\t\\label{matrnewUQg}\n\\eea\nFrom  (\\ref{matrnewUQg}) it is obvious that up to the normalization factor the  dequantizer \n$\\hat U_\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ is the projector of the considered state  \n$\\hat\\rho(t)$ onto the pure state $|\\Psi_{\\mbf{q,p}}\\rangle$, i.e. \n$\\hat U_\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)=(2\\pi\\hbar)^{-3}|\\Psi_{\\mbf{q,p}}\\rangle\\langle\\Psi_{\\mbf{q,p}}|$,\nwhere  we have up to the phase factor in the position representation:\n\\be\n\\langle\\mbf q'|\\Psi_{\\mbf{q,p}}\\rangle=\n\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{3\/4}\n\\exp\\bigg[-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q')^2\n-\\frac{\\rmi}{\\hbar}\\mbf p(\\mbf q-\\mbf q')\n+\\frac{\\rmi e}{\\hbar c}\\mbf q'\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q',t)\n\\bigg].\n\\ee\nCalculations of the matrix element of the quantizer for the function $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\ngive rise to the expression\n\\bea\n\\langle\\mbf q_1|\\hat D_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_2\\rangle&=&\n\\int\\frac{d^3v}{(2\\pi\\hbar)^3}\\exp\\bigg[\\frac{\\lambda(\\mbf q_2-\\mbf q_1)^2}{4\\hbar}\n+\\frac{\\mbf v^2}{4\\hbar\\lambda}-\\frac{\\rmi\\mbf v}{2\\hbar}(2\\mbf q-\\mbf q_1-\\mbf q_2)\n-\\frac{\\rmi\\mbf p}{\\hbar}(\\mbf q_2-\\mbf q_1) \\nonumber \\\\[3mm]\n&-&\\frac{\\rmi e}{\\hbar c}\\mbf q_2\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_2,t)\n+\\frac{\\rmi e}{\\hbar c}\\mbf q_1\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_1,t)\n\\bigg].\n\t\t\t\\label{matrnewDg}\n\\eea\nWhen using quantizer (\\ref{matrnewDg}), it is assumed that\n$\\mathfrak{Q}_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)\\in S(\\mathbb{R}^6)$,\nwhere $S(\\mathbb{R}^6)$ is a Schwartz space, and first we take the integral over $d^3q$,\nand after that the integrals over $d^3p$ and $d^3v$ are taken.\nWith the same stipulation, the explicit form of the operator $\\hat D_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)$\ncan be written as:\n\\bea\t\t\t\\label{newDQgint}\n\\hat D_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)&=&\\exp\\left[\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]\n\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^3}\n\\exp\\left[\\frac{\\lambda \\mbf u^2}{4\\hbar}+\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf{up}+\\mbf{vq})\\right]\n\\nonumber \\\\[3mm]\n&\\times&\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\hat{\\mbf P}\n+\\mbf v\\hat{\\mbf q}\\right]\\right\\}\\exp\\left[-\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right].\n\\eea\n\nKnowing the quantizers and dequantizers for functions\n$\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$ and $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\none can find the evolution equations for them.\nSince $\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$ and $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\ndo not depend on the gauge, then their evolution equations must also be gauge-independent.\n\n\n\\section{\\label{Art08Section5}Conclusion}\nIn conclusion, I point out the main results of the paper.\nThe gauge-independent Husimi function ($Q-$function) of states of charged quantum \nparticles in the electro-magnetic field was introduced using the gauge-independent\nStratonovich-Wigner function, \nthe corresponding dequantizer and quantizer operators transforming the density matrix\nof state to the such Husimi function and vice versa were found explicitly,\nand the evolution equation for such function was derived.\n\nAlso own non-Stratonovich gauge-independent Wigner function  and its \nHusimi function were suggested and their dequantizers and quantizers were obtained.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, dialogue systems based on KGs have become increasingly popular because of their wide range of applications from hotel bookings, customer-care to voice assistant services. Such dialogue systems can be realized using both goal and non-goal oriented methods. Whereas the former one is employed for carrying out a particular task, the latter is focused on performing natural (\"chit-chat\") dialogues. Both types of dialogue system can be implemented using a generative approach. In a generative dialogue system, the response is generated (usually word by word) from the domain vocabulary given a natural language user query, along with the previous dialogue context. Such systems can benefit from the integration of additional world knowledge~\\cite{eric2017key}. In particular, knowledge graphs, which are an abstraction of real world knowledge, have been shown to be useful for this purpose. Information of the real world can be stored in a KG in a structured (Resource Description Framework (RDF) triple, e.g., $<subject, relation, object>$) and abstract way (Paris is the capital city of France and be presented in $<Paris, capital \\: city, France>$).\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{images\/intro_graph.png}\n    \\caption{Example of a knowledge grounded conversation.}\n    \\label{fig:kg_convo}\n\\end{figure}\nKG based question answering (KGQA) is already a well-researched topic~\\cite{chakraborty2019introduction}. However, generative dialogue systems with integrated KGs have only been explored more recently~\\cite{chaudhuri2019using,madotto2018mem2seq,eric2017key}.\nTo model the response using the KG, all current methods assume that the entity in the input query or a sub-graph of the whole KG, which can be used to generate the answer, is already known~\\cite{madotto2018mem2seq,wu2019global}.\nThis assumption makes it difficult to scale such systems to real-world scenarios, because the task of extracting sub-graphs or, alternatively, performing entity linking in large knowledge graphs is non-trivial~\\cite{rosales2018should}. An example of a knowledge graph based dialogue system is shown in Figure~\\ref{fig:kg_convo}. In order to generate the response \\textit{James Cameron is the director}, the system has to link the entity mentioned in the question in the first turn i.e.~\\textit{Titanic}, and identify the relation in the KG connecting the entities \\textit{Titanic} with \\textit{James Cameron}, namely \\textit{directed by}. Additionally, to obtain a natural dialogue system, it should also reply with coherent responses (eg. \"James Cameron is the director\") and should be able to handle small-talk such as greetings, humour etc. Furthermore, in order to perform multi-turn dialogues, the system should also be able to perform co-reference resolution and connect the pronoun (\\textit{he}) in the second question with \\textit{James Cameron}. \n\nIn order to tackle these research challenges, we model the dialogue generation process by jointly learning the entity and relation information during the dialogue generation process using a pre-trained BERT model in an end-to-end manner. The model's response generation is designed to learn to predict relation(s) from the input KG instead of the actual object(s) (intermediate representation).  Additionally, a graph Laplacian based method is used to encode the input sub-graph and use it for the final decoding process.\n\nExperimental results suggest that the proposed method improves upon previous state-of-the-art approaches for both goal and non-goal oriented dialogues. Our code is publicly available on Github~\\footnote{\\url{https:\/\/github.com\/SmartDataAnalytics\/kgirnet\/}}. Overall, the contributions of this paper are as follows:\n\n\n\n\n\n\n\\noindent \n\\begin{itemize}\n\t\\item A novel approach, leveraging the knowledge graph elements (entities and relations) in the questions along with pre-trained transformers, which helps in generating suitable knowledge grounded responses. \n\t\\item We have also additionally encoded the sub-graph structure of the entity of the input query with a Graph Laplacian, which is traditionally used in graph neural networks. This novel decoding method further improves performance.\n\n\n\t\\item An extensive evaluation and ablation study of the proposed model on two datasets requiring grounded KG knowledge: an in-car dialogue dataset\n\tand soccer dialogues\n\tfor goal and non-goal oriented setting, respectively.\n\tEvaluation results show that the proposed model produces improved knowledge grounded responses compared to other state-of-the-art dialogue systems w.r.t.~automated metrics, and human-evaluation for both goal and non-goal oriented dialogues.\n\n\\end{itemize}\n\n\\section{Model Description}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/model_diagram.png}\n    \\caption{KGIRNet model diagram.}\n    \\vspace{0.1mm}\n    \\label{fig:model_diagram}\n\\end{figure*}\n\nWe aim to solve the problem of answer generation in a dialogue using a KG as defined below.\n\n\n\\begin{definition}[Knowledge Graph]\nWithin the scope of this paper, we define a \\emph{knowledge graph} $KG$ as a labelled, undirected multi-graph consisting of a set $V$ of nodes and a set $E$ of edges between them. There exists a function, $f_l$ that maps the nodes and vertices of a graph to a string. The neighborhood of a node of radius $k$ (or $k$-hop) is the set of nodes at a distance equal to or less than $k$. \n\\label{def:kg}\n\\end{definition}\nThis definition is sufficiently generic to be applicable to knowledge graphs based on RDF\\footnote{\\url{https:\/\/www.w3.org\/RDF\/}} (Resource Description Framework) as well as property graphs~\\cite{gubichev2014graph}. The vertices $V$ of the $KG$ represent entities $e \\in V$, while the edges represent the relationships between those entities. A fact is an ordered triple consisting of an entity $e$ (or subject $s$), an object $o$ and the relation $r$ (a.k.a. predicate $p$) between them, denoted by ($s$, $p$, $o$).\n\nThe proposed model for dialogue generation, which we call KGIRNet, is quintessentially a sequence-to-sequence based model with a pre-trained transformer serving as its input as illustrated in Figure~\\ref{fig:model_diagram}. In contrast to previous works, we introduce an intermediate query representation using the relation information, for training. We also employ a Graph Laplacian based method for encoding the input sub-graph of the KG that aids in predicting the correct relation(s) as well as filter out irrelevant KG elements.\nOur approach consists of the following steps for which more details are provided below:\nFirstly, we encode the input query $q$ with a pre-trained BERT model (Section~\\ref{sec:bert}). Next we detect a core entity $e$ occurring in $q$ (Section~\\ref{sec:entitydetection}). Input query $q$ and generated output is appended to the dialogue context in every utterance. The input query, encoded using the BERT model is then passed through an LSTM encoder to get an encoded representation (Section~\\ref{query_enc}). This encoded representation is passed onto another LSTM decoder (Section~\\ref{dec}), which outputs a probability distribution for the output tokens at every time-step. Additionally, the model also extracts the $k$-hop neighbourhood of $e$ in the KG and encodes it using graph based encoding (Section~\\ref{sub_graph}) and perform a Hadamard product with token probability distribution from the decoder. The decoding process stops when it encounters a special token, $<EOS>$ (end of sentence). Dotted lines in the model diagram represent operations performed at a different time-step $t$ in the decoding process where solid lines are performed once for each utterance or input query. \n\nIn this work, we define complex questions as questions which require multiple relations to answer the given question. For example, for the following query: \"\\textit{please tell me the location, time and the parties that are attending my meeting}\", the model needs to use 3 relations from the KG for answering, namely location, time and parties. The answer given by the model could be : \"\\textit{you have meeting scheduled on friday at 10am with boss in conference\\_room\\_102 to go\\_over\\_budget}\". The model is able to retrieve  important relation information from the KG during decoding. However, the model is not able to handle questions which go beyond the usage of explicitly stored relations and require inference capabilities .\n\n\\subsection{Query Encoding}\n\\label{sec:bert}\n\nBERT is a pre-trained multi-layer, bi-directional transformer~\\cite{vaswani2017attention} model proposed in~\\cite{devlin2019bert}. It is trained on unlabelled data for two-phased objective: masked language model and next sentence prediction.  For encoding any text, special tokens [CLS] and [SEP] are inserted at the beginning and the end of the text, respectively. In the case of KGIRNet, the input query  $q = (q_1, q_2,...q_n)$ at turn $t_d$ in the dialogue, along with the context up to turn $t_{d}-1$ is first encoded using this pre-trained BERT model which produces hidden states ($T_1, T_2....T_n$) for each token and an aggregated hidden state representation $C$ for the [CLS] (first) token.\nWe encode the whole query $q$ along with the context, concatenated with a special token, $<EOU>$ (end of utterance).\n\n\\subsection{Entity Detection}\n\\label{sec:entitydetection}\n\nThe aggregated hidden representation from the BERT model $C$ is passed to a fully connected hidden layer to predict the entity $e_{inp} \\in V$  in the input question as given by\n\\begin{equation}\n    e_{inp} = softmax(w_{ent}C + b_{ent})\n\\end{equation}\n\n\\noindent Where, $w_{ent}$ and $b_{ent}$ are the parameters of the fully connected hidden layer.\n\n\n\\subsection{Input Query Encoder}\n\\label{query_enc}\nThe hidden state representations ($T_1, T_2....T_n$) of the input query $q$ (and dialogue context) using BERT is further encoded using an LSTM \\cite{hochreiter1997long} encoder which produces a final hidden state at the $n$-th time-step given by\n\n\\begin{equation}\n    h^e_n = f_{enc}(T_n, h^e_{n-1})\n\\end{equation}\n\n\\noindent $f_{enc}$ is a recurrent function and $T_n$ is the hidden state for the input token $q_n$ from BERT.\n\n\\noindent The final representation of the encoder response is a representation at every $n$ denoted by\n\n\\begin{equation}\n    H_{e} = (h^e_0, h^e_1....h^e_N)\n\\end{equation}\n\n\n\\subsection{Intermediate Representation}\n\\label{graph_ir}\n\nAs an intermediate response, we let the model learn the relation or edge label(s) required to answer the question, instead of the actual object label(s). In order to do this, we additionally incorporated the relation labels obtained by applying the label function $f_l$ to all edges in the KG into the output vocabulary set. If the output vocabulary size for a vanilla sequence-to-sequence model is $v_{o}$, the total output vocabulary size becomes $v_{od}$ which is the sum of $v_{o}$ and $v_{kg}$. The latter being the labels from applying the $f_l$ to all edges (or relations) in the KG.\n\nFor example, if in a certain response, a token corresponds to an object label $o_l$ (obtained by applying $f_l$ to $o$) in the fact $(e, r, o)$, the token is replaced with a KG token $v_{kg}$ corresponding to the edge or relation label $r_l$ of $r\\in E$ in the KG.\nDuring training, the decoder would see the string obtained by applying $f_l$ to the edge between the entities \\textit{Titanic} and \\textit{James Cameron}, denoted here as \\textit{r:directedBy}. Hence, it will try to learn the relation instead of the actual object. This makes the system more generic and KG aware, and easily scalable to new facts and domains.\n\nDuring evaluation, when the decoder generates a token from $v_{kg}$, a KG lookup is done to decode the label $o_l$ of the node $o \\in V$ in the KG ($V$ being the set of nodes or vertices in the KG). This is generally done using a SPARQL query.\n\n\\subsection{Decoding Process}\n\\label{dec}\nThe decoding process generates an output token at every time-step $t$ in the response generation process. It gets as input the encoded response $H_{e}$ and also the KG distribution from the graph encoding process as explained later. The decoder is also a LSTM, which is initialized with the encoder last hidden states and the first token used as input to it is a special token, $<SOS>$ (start of sentence). \nThe decoder hidden states are similar to that of the encoder as given by the recurrent function $f_{dec}$\n\n\\begin{equation}\n    h^d_n = f_{dec}(w_{dec}, h^d_{n-1})\n\\end{equation}\n\n\\noindent This hidden state is used to compute an attention over all the hidden states of the encoder following \\cite{luong2015effective}, as given by \n\n\\begin{equation}\n    \\alpha_t = softmax(W_s(tanh(W_c[H_{e}; h^d_t]))\n\\end{equation}\n\n\\noindent Where, $W_c$ and $W_s$ are the weights of the attention model.\nThe final weighted context representation is given by  \n\\begin{equation}\n    \\Tilde{h_t} = \\sum_t \\alpha_t h_t\n\\end{equation}\n\n \\noindent This representation is concatenated (represented by $;$) with the hidden states of the decoder to generate an output from the vocabulary with size $v_{od}$.\n \n \\noindent The output vocab distribution from the decoder is given by\n \n \\begin{equation}\n     O_{dec} = W_o([{h_t; \\Tilde{h^d_t}}])\n \\end{equation}\n \n \\noindent In the above equation, $W_o$ are the output weights with dimension $\\mathbf{R}^{h_{dim}Xv_{od}}$. $h_{dim}$ being the dimension of the hidden layer of the decoder LSTM. The total loss is the sum of the vocabulary loss and the entity detection loss.\n Finally, we use beam-search \\cite{tillmann2003word} during the the decoding method. \n \n\\subsection{Sub-Graph Encoding}\n \\label{sub_graph}\n\n In order to limit the KGIRNet model to predict only from those relations which are connected to the input entity predicted from step~\\ref{sec:entitydetection}, we encode the sub-graph along with its labels and use it in the final decoding process while evaluating.\n \n\\noindent The $k$-hop sub-graph of the input entity is encoded using Graph Laplacian \\cite{kipf2016semi} given by\n\n\\begin{equation}\n    G_{enc} = D^{-1}\\Tilde{A}f_{in}\n    \\label{graph_spec}\n\\end{equation}\n\n\\noindent Where, $\\Tilde{A} = A+I$. $A$ being the adjacency matrix, $I$ is the identity matrix and D is the degree matrix. $f_{in}$ is a feature representation of the vertices and edges in the input graph. \n$G_{enc}$ is a vector with dimensions $\\mathbf{R}^{ik}$ corresponding to the total number of nodes and edges in the $k$-hop sub-graph of $e$. An example of the sub-graph encoding mechanism is shown in Figure \\ref{fig:graph_lapl}.\n \n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{images\/KGIRNET-graphLapl.png}\n    \\caption{Sub-Graph Encoding using Graph Laplacian.}\n    \\vspace{-0.1mm}\n    \\label{fig:graph_lapl}\n\\end{figure*}\n\n\\noindent The final vocabulary distribution $O_{f} \\in R^{v_{od}}$ is a Hadamard product of this graph vector and the vocabulary distribution output from the decoder. \n\n\\begin{equation}\n    O_{f} = O_{dec} \\odot G_{enc}\n\\end{equation}\n\n\\noindent This step essentially helps the model to give additional importance to the relations connected at k-hop based on its similarity with the query also to filter(mask) out relations from the response which are not connected to it. For the query in \\ref{fig:graph_lapl} \\textit{who is the director of Avatar} and {how was it rated ?} The graph laplacian based encoding method using only relation labels already gives higher scores for the relations directed\\_by and rating, which are required by the question. This vector when multiplied with the final output vocabulary helps in better relation learning.\n\n\n\n\\section{Experimental Setup}\n\n\\subsection{Datasets}\n\nAvailable datasets for knowledge grounded conversations are the \\textit{in-car dialogue} data as proposed by~\\cite{eric2017key} and \\textit{soccer dialogues} over football club and national teams using a knowledge graph~\\cite{chaudhuri2019using}. The former contains dialogues for assisting the driver of a car with data related to weather, schedules and navigation, in a goal-oriented setting.\nThe soccer dataset contains non-goal oriented dialogues about questions on different soccer teams along with a knowledge graph consisting of facts extracted from Wikipedia about the respective clubs and national soccer teams. Both the datasets were collected using Amazon Mechanical Turk (AMT) by the respective authors~\\cite{eric2017key,chaudhuri2019using}. The statistics of the datasets are provided in Table ~\\ref{tab:stats}. As observed, the number of dialogues for both the dataset is low.\n\n\nTo perform a KG grounded dialogue as in KG based question-answering~\\cite{diefenbach2018core}, it is important to annotate the dialogues with KG information such as the entities and relations, the dialogues are about. Such information were missing in the soccer dataset, hence we have semi-automatically annotated them with the input entity $e_{inp}$ in the query $q$ and the relations in the $k$-hop sub-graph of the input entity required to answer $q$. For the domain of in-car it was possible to automatically extract the entity and relation information from the dialogues and input local KG snippets.\n\n\\vspace{-.8cm}\n\\begin{table}[]\n\\centering\n\\caption{Dataset statistics.}\n\\begin{tabular}{l|c|c|c|c|c|c}\n\\toprule\n     & \\multicolumn{3}{c|}{\\textbf{In-car dialogues}} & \\multicolumn{3}{c}{\\textbf{Soccer dialogues}} \\\\\\toprule\nNumber of triples, entity, relations &   \\multicolumn{3}{c|}{8561, 271, 36}    &  \\multicolumn{3}{c}{4301, 932, 30}\\\\\\midrule\n     & \\textbf{Train}     & \\textbf{Validation}      & \\textbf{Test}     & \\textbf{Train}     & \\textbf{Validation}     & \\textbf{Test}      \\\\\\midrule\nNumber of dialogues &     2011      &        242        &    256      &     1328      &     149     &      348          \\\\\nNumber of utterances &      5528     &        657        &    709      &    6523       &     737     &       1727         \\\\\nKG-grounded questions (\\%) &    44.95       &       33.94         &     43.84     &     6.53      &   4.61       &  3.88  \\\\\n\\bottomrule\n\\end{tabular}\n\\label{tab:stats}\n\\end{table}\n\\vspace{-.6cm}\n\\subsection{Evaluation Metrics}\n\nWe evaluate the models using the standard evaluation metrics BLEU~\\cite{papineni2002bleu} and Entity F1 scores as used in the discussed state-of-the-art models. However, unlike~\\cite{madotto2018mem2seq} we use Entity F1 scores based on the nodes $V$ of the KG, as inspired by previous works on KG based question answering~\\cite{Bordes2015LargescaleSQ}. Additionally, we use METEOR~\\cite{banerjee2005meteor} because it correlates the most with human judgement scores~\\cite{sharma2017relevance}.\n\n\n\\subsection{Model Settings}\n\\begin{table*}[ht]\n    \\caption{Evaluation on goal and non-goal oriented dialogues.}\n    \\centering \n   \\vspace{.1cm}\n    \\def1{1.2}\n\\begin{adjustbox}{width=0.8\\textwidth}\n\\begin{tabular} {c c|c|c|c|c|c|c}\n\\toprule\n\\multirow{2}{*}{\\textbf{Models}} &  \\multicolumn{3}{c}{\\textbf{In-Car Dialogues}} & \n\\multicolumn{3}{c}{\\textbf{Soccer Dialogues}} & Inference time\\\\\n\\cline{2-8}\n\n & \\textit{BLEU} & \\textit{Entity F1} & \\textit{METEOR} &\\textit{BLEU} & \\textit{Entity F1} & \\textit{METEOR} & \\textit{utterances\/sec}\\\\\n\\hline\nSeq2Seq & 4.96 & 10.70  & 21.20 & 1.37 & 0.0 & 7.8 & 133\\\\\nMem2Seq\\cite{madotto2018mem2seq} & 9.43 & 24.60 & 28.80 & 0.82 & 04.95 & 7.8 & 88 \\\\\nGLMP\\cite{wu2019global} & 9.15 & 21.28 & 29.10 & 0.43 & 22.40 & 7.7 & 136\\\\\nTransformer\\cite{vaswani2017attention} & 8.27 & 24.79 & 29.06 &0.45 & 0.0 & 6.7 & 7\\\\\nDialoGPT\\cite{zhang2019dialogpt} & 7.35 & 21.36 & 20.50 & 0.76 & 0.0 & 5.5 & 2\\\\\nKG-Copy\\cite{chaudhuri2019using} & - & - & - & \\textbf{1.93} & 03.17 & \\textbf{10.89} & 262\\\\\n\\midrule\n\nKGIRNet & \\textbf{11.76} & \\textbf{32.67} & \\textbf{30.02} & 1.51 & \\textbf{34.33} & 8.24 & 37\\\\\n\\bottomrule\n\n\\end{tabular}\n    \\label{tab:eval}\n    \\end{adjustbox}\n\\end{table*}\n\nFor entity detection we used a fully connected layer on top of CNN-based architecture. Size of the hidden layer in the fully connected part is 500 and a dropout value of 0.1 is used and ReLU as the activation function. In the CNN part we have used 300 filters with kernel size 3, 4 and 5. We use BERT-base-uncased model for encoding the input query. The encoder and decoder is modelled using an LSTM (long short term memory) with a hidden size of 256 and the KGIRNet model is trained with a batch size of 20. The learning rate of the used encoder is 1e-4. For the decoder we select a learning rate of 1e-3. We use the Adam optimizer to optimize the weights of the neural networks. For all of our experiments we use a sub-graph size of k=2. For calculating $f_{in}$ as in Equation~\\ref{graph_spec}, we use averaged word embedding similarity values between the query and the labels of the sub-graph elements. The similarity function used in this case is cosine similarity. We save the model with best validation Entity f1 score.\n\\vspace{-.8cm}\n\n\\begin{table}[ht]\n\\caption{Relation Linking Accuracy on SQB \\cite{wu-etal-2019-learning-representation} Dataset.}\n    \\centering \n    \n   \n\\begin{adjustbox}{width=0.6\\textwidth}\n\\begin{tabular} {p{20mm} | p{40mm} | p{20mm}}\n\\toprule\n\\textbf{Method} & \\textbf{Model} & \\textbf{Accuracy}\\\\\n\\midrule\n\\textbf{Supervised} & {Bi-LSTM \\cite{mohammed-etal-2018-strong}} & {38.5}\\\\ \n{} & {HR-LSTM \\cite{yu-etal-2017-improved}} & {64.3} \\\\ \n\\midrule\n\\textbf{Unsupervised} &{Embedding Similarity} & {60.1} \\\\ \n\n{} & {Graph Laplacian (this work)} & \\textbf{69.7}  \\\\ \n\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\n    \\label{tab:sq_rl}\n\\end{table}\n\\vspace{0.02cm}\n\\section{Results}\n\\label{results}\nIn this section, we summarize the results from our experiments. We evaluate our proposed model KGIRNet against the current state-of-the-art systems for KG based dialogues; namely Mem2Seq~\\cite{madotto2018mem2seq}, GLMP~\\cite{wu2019global}, and KG-Copy~\\cite{chaudhuri2019using}\\footnote{KG-Copy reports on a subset of the in-car testset hence it is not reported here}. To include a baseline method, the results from a vanilla Seq2Seq model using an LSTM encoder-decoder are also reported in Table~\\ref{tab:eval}, along with a vanilla transformer~\\cite{vaswani2017attention} and a pre-trained GPT-2 model, DialoGPT \\cite{zhang2019dialogpt} fine-tuned individually on both the datasets.\n\nWe observe that KGIRNet outperforms other approaches for both goal (in-car) and non-goal oriented (soccer) dialogues except for BLEU and METEOR scores on the soccer dataset.\nThe effect of knowledge groundedness is particularly visible in the case of soccer dialogues, where most models (except GLMP) produces feeble knowledge grounded responses. The Entity F1 score used here is adapted from \\cite{Bordes2015LargescaleSQ} and is defined as the average of F1-scores of the set of predicted objects, for all the questions in the test set.\n\nIn addition to evaluating dialogues, we also evaluate the proposed graph laplacian based relation learning module for the task of knowldge-graph based relation linking. Although, it is a well-researched topic and some systems claim to have solved the problem \\cite{petrochuk-zettlemoyer-2018-simplequestions}, but such systems are not able to handle relations which are not present in the training data \\cite{wu-etal-2019-learning-representation}. The latter also proposed a new, balanced dataset (SQB) for simple question answering which has same proportions of seen and unseen relations in the test or evaluation set. We have evaluated our unsupervised graph laplacian based method for relation linking on the SQB dataset against supervised methods namely Bi-LSTM \\cite{mohammed-etal-2018-strong}, hr-bilstm \\cite{yu-etal-2017-improved} and unsupervised method such as text embedding based similarity between the query and the relations connected to the subject entity with 1-hop. The results are reported in Table \\ref{tab:sq_rl}. As observed, \nGraph Laplacian performs better wrt. supervised methods on unseen relations and also better than shallow embedding based similarity. This is one of the motivation for using this simple method during KGIRNet's decoding process. We run all the inference on a setup with 32GB of RAM and 8GB of VGA and a Bandwidth of 256.3 GB\/s. \t\n\n\n\\section{Discussion}\n\\label{discussion}\n\nFor in-car dialogues, we train and evaluate on queries which require knowledge from the KG, hence we omit the scheduling conversations where the driver asks the car to schedule a meeting\/conference.\nIn order to get the knowledge graph candidate triples for all the other models (Mem2Seq and GLMP), we provide them with the 2-hop sub-graph of the correct input entity instead of the local knowledge as used in the respective models; this, represents a scenario closer to a real-world KG grounded conversation.\nFor the in-car dataset, the human baseline BLEU score as reported in~\\cite{eric2017key} is 13.5 (the KGIRNet score is 11.76).\nThe BLEU scores for soccer are low because non-goal oriented dialogues are more complicated to model due to large vocabulary sizes (more than 3 times the vocabulary size of the of in-car dataset). Also in the case of soccer dialogues, number of factual conversation is low (4\\%) compared to the total number of utterances and also the conversations are more generic and repetitive in nature.\n\\vspace{-.8cm}\n\\begin{table*}[ht]\n \\caption{Analyzing sample predictions.}\n    \\centering \n    \\vspace{.1cm}\n   \n\\begin{adjustbox}{width=\\textwidth}\n\\begin{tabular} {p{30mm}|p{25mm}|p{30mm}|p{35mm}|p{30mm}}\n\n\\toprule\n\\multirow{2}{*}{\\textbf{Input Query}} &\n\\multirow{2}{*}{\\textbf{True Response}} &\n\\multicolumn{3}{c}{\\textbf{Predicted Responses}} \\\\\n\\cline{3-5}\n\n & & \\textbf{GLMP} & \\textbf{Mem2Seq} & \\textbf{KGIRNet} \\\\%& \\textit{True} & \\textit{Predicted} \\\\\n\\hline\n{\\textbf{(S)} who is currently coaching bvb dortmund ? } & {lucien favre} & \\textit{the is the coach} & \\textit{yes , they have a good}  & \\textit{lucien favre is the coach of bvb dortmund} \\\\\n\\midrule\n{\\textbf{(S)} when did italy last win the world cup ?} & 2006 & \\textit{italy won the world cup in 2006} & \\textit{i think they have a good team} & \\textit{italy won the world cup in 2006} \\\\\n\\midrule\n\\midrule\n\n{\\textbf{(C)} what time is my doctorappointment?} & {your doctorappointment is scheduled for friday at 11am} & \\textit{your next is is at 1pm at 7pm} & \\textit{your doctorappointment is at 1pm} & \\textit{your doctorappointment is on friday at 11am} \\\\\n\\midrule\n\\textbf{(C)} i need gas &\nvalero is 4\\_miles away & \\textit{there is a valero away} & \\textit{chevron is gas\\_station away chevron is at away} & \\textit{there is a valero nearby} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n   \n    \\label{tab:resp_analysis}\n\\end{table*}\n\\vspace{-.8cm}\n\\subsection{Human Evaluation}\nWe  perform  a  human-based  evaluation  on  the  whole  test  dataset  of  the generated responses from the KGIRNet model and its closest competitors, i.e. Mem2Seq, GLMP and DialoGPT models. We asked 3 internal independent annotators who are not the authors of this paper (1 from CS and 2 from non-CS background) to rate the quality of the generated responses between 1-5 with respect to the dialogue context (higher is better). Note that for each dataset, we provide the annotators with 4 output files in CSV format (containing the predictions of each model) and the knowledge graph in RDF format. Each of the CSV files contains data in 5 columns: question, original response, predicted response, grammatical correctness, similarity between original and predicted response. In the provided files, the 4th (grammatically correctness) and 5th (similarity between original and predicted response) columns are empty and we ask the annotators to fill them with values (within a range of 1-5) according to their judgement. The measures requested to evaluate upon are correctness (Corr.) and human-like (human) answer generation capabilities. Correctness is measured by comparing the response to the correct answer object. Reference responses are provided besides the system generated response in order to check the factual questions. The results are reported in Table~\\ref{tab:humaneval}. Cohen's Kappa of the annotations is 0.55.\n\n\\subsection{Ablation Study}\nAs an ablation study we train a sequence-to-sequence model with pre-trained fasttext embeddings as input (S2S), and the same model with pre-trained BERT as input embedding (S2S\\_BERT). Both these models do not have any information about the structure of the underlying knowledge graph. Secondly, we try to understand how much the intermediate representation aids to the model, so we train a model (KGIRNet\\_NB) with fasttext embeddings as input instead of BERT along with intermediate relation representation. Thirdly, we train a model with pre-trained BERT but without the haddamard product of the encoded sub-graph and the final output vocabulary distribution from Step~\\ref{sub_graph}. This model is denoted as KGIRNet\\_NS. As observed, models that are devoid of any KG structure has very low Entity F1 scores, which is more observable in the case of soccer dialogues since the knowledge grounded queries are very low, so the model is not able to learn any fact(s) from the dataset. The proposed intermediate relation learning along with Graph Laplacian based sub-graph encoding technique observably assists in producing better knowledge grounded responses in both the domains; although, the latter aids in more knowledge groundedness in the domain of soccer dialogues (higher Entity F1 scores). We also did an ablation study on the entity detection accuracy of the end-to-end KGIRNet model in the domain of in-car and compared it with a standalone Convolutional neural network (CNN) model which predicts the entity from the input query, the accuracies are 79.69 \\% and 77.29\\% respectively. \n\n\\vspace{-.9cm}\n\\begin{table}[ht]\n\\caption{In-depth evaluation of KGIRNet model.}\n   \n        \\begin{subtable}{.48\\linewidth}\n    \\begin{tabular} {c|c|c|c|c}\n\\toprule\n\\multirow{2}{*}{\\textbf{Models}} &  \\multicolumn{2}{c|}{\\textbf{In-Car}} & \n\\multicolumn{2}{c}{\\textbf{Soccer}}\\\\\n\\cline{2-5}\n\n & \\textit{Corr.} & \\textit{Human} & \\textit{Corr.} & \\textit{Human} \\\\\n\\hline\nMem2Seq & 3.09 & 3.70 & 1.14 & 3.48  \\\\\nGLMP & 3.01 & 3.88 & 1.10 & 2.17 \\\\\nDialoGPT  & 2.32  & 3.43 & 1.32 & 3.88  \\\\\n\\midrule\nKGIRNet & 3.60 & 4.42 & 1.59 & 3.78 \\\\\n\n \\bottomrule\n\\end{tabular}\n    \\caption{Human evaluation.}\n    \\label{tab:humaneval}\n    \\end{subtable}\n   \n    \\begin{subtable}{.52\\linewidth}\n   \n   \n\\begin{adjustbox}{width=\\textwidth}\n\n\\begin{tabular} {lc|c|c|c}\n\\toprule\n\\multirow{2}{*}{\\textbf{Models}} &  \\multicolumn{2}{c|}{\\textbf{In-Car Dialogues}} & \n\\multicolumn{2}{c}{\\textbf{Soccer Dialogues}}\\\\\n\\cline{2-5}\n\n & \\textit{BLEU} & \\textit{EntityF1} & \\textit{BLEU} & \\textit{EntityF1} \\\\\n\\hline\nS2S & 4.96 & 10.70 & 1.49 & 0.0 \\\\\nS2S\\_BERT & 7.53 & 09.10 & 1.44 & 0.0 \\\\\nKGIRNet\\_NB & 9.52 & 29.03 & 0.91 & 29.85 \\\\\nKGIRNet\\_NS & 11.40 & 33.03 & 1.05 & 28.35 \\\\\n\\midrule\nKGIRNet & 11.76 & 32.67 & 1.51 & 34.32 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n    \\caption{Ablation study.}\n    \\label{tab:ablation}\n    \\end{subtable}\n\\vspace{0.1mm}    \n\n\\end{table}\n\\vspace{-.9cm} \n\n\\subsection{Qualitative Analysis}\nThe responses generated by KGIRNet are analyzed in this section. Responses from some of the discussed models along with the input query are provided in Table~\\ref{tab:resp_analysis} \\footnote{In the table \\textbf{(S)} and \\textbf{(C)} refers to example from Soccer and In-car dataset respectively.}. We compare the results with two other state-of-the-art models with the closest evaluation scores, namely Mem2Seq and GLMP. The first two responses are for soccer dialogues, while the latter two are for in-car setting. We inspect that  qualitatively the proposed KGIRNet model produces better knowledge grounded and coherent responses in both the settings. In the case of soccer dialogues, predicting single relation in the response is sufficient, while for the case of in-car dialogues, responses can require multiple relation identification. KGIRNet model is able to handle such multiple relations as well (e.g., $r$:date friday and $r$:time 11am for the third utterance).\n\n\\vspace{-.9cm}\n\\begin{table}[ht]\n\\caption{Analyzing fact-fullness of KGIRNet.}\n    \\centering \n    \\vspace{.1cm}\n   \n\n\\begin{adjustbox}{width=1\\textwidth}\n\\begin{tabular} {p{30mm}|p{28mm}|p{27mm}|p{35mm}}\n\n\\toprule\n{\\textbf{Input Query}} &  \\textbf{True Response} & \n\\textbf{Predicted Response} & \\textbf{Intermediate Response}\\\\\n\\midrule\n\n{who is senegal captain ?} & {cheikhou kouyate} & \\textit{sadio mane is the captain of senegal} & \\textit{r:captain is the captain of @entity} \\\\\n\\hline\n{who is the goalie for iceland ?} & hannes halldorsson & \\textit{runar alex runarsson} & \\textit{$r$\\:goalkeeper} \\\\\n\\bottomrule\n\\end{tabular}\n    \\label{tab:intermed}\n\\end{adjustbox}\n\n\\end{table}\n\\vspace{-.6cm}\nTo understand more about the advantages of KG grounded generation process, consider the examples from the soccer dataset below in Table~\\ref{tab:intermed}, in which the model has failed. We have also included the intermediate responses that are generated from the decoder before the KG lookup phase for the reference.\nIn the first case, while constructing the KG, the captain was \\textit{Sadio Mane}, whereas the during the dialogue collection phase the probable captain was \\textit{Checikhou Kouyate} (as per the annotator). These errors on temporal information can easily be tackled using Knowledge Grounded dialogue systems, where the system can potentially reflect current information with updated real world knowledge. For the second case, the current main goalkeeper for the Iceland national football team is \\textit{Hannes Halld\u00f3rsson}; however, \\textit{R\u00fanar Alex R\u00fanarsson} is also among the 3 goalkeepers for the team and the model has no additional information to disambiguate between them to produce the same response as the human annotator. KGIRNet is also able to infer conversational context implicitly in dialogues with the system (Table~\\ref{tab:kgirnet_convo}).\n\\vspace{-.9cm}\n\\begin{table}[ht]\n\\caption{A conversation with KGIRNet.}\n    \\centering \n    \\vspace{.2cm}\n   \n\\begin{adjustbox}{width=0.8\\textwidth}\n\\begin{tabular} {p{18mm} p{100mm}}\n\\toprule\n\\textbf{User:} & {what s the weather forecast for today and tomorrow ?}\\\\\n\\midrule\n\\textbf{KGIRNet:} & {what city do you want the weather for}\\\\ \n\\midrule\n\\textbf{User:} & {los angeles} \\\\ \n\\midrule\n\\textbf{KGIRNet:} & {it will be 40f - 60f on monday warm on tuesday windy on wednesday hot on thursday} \\\\ \n\\midrule\n\\textbf{User:} & {thanks}  \\\\ \n\\midrule\n\\textbf{KGIRNet:} & {you are welcome} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n    \n    \\label{tab:kgirnet_convo}\n\\end{table}\n\\vspace{-.6cm}\n\\subsection{Error Analysis}\n\nTo analyze more on the shortcomings of our proposed KGIRNet, we stated some sample erroneous responses from the model in Table~\\ref{tab:errors}. The first three are from soccer and the rest in-car dialogues. The model often produces generic responses for soccer as in the first example. Our model is only able to handle simple factoid queries and not able to answer queries requiring comparative reasoning such as \\textit{best player}, \\textit{youngest player} etc, as observable from the second and third cases. In the second case, the model answers with the coach's name and when asked for the best player for Argentina and it replies with the name of an Uruguayan player in the third case. The model also produces generic relations like coach name, number of world cup championship as response when any such keyword is present in the input. For the in-car dataset, the replies are less generic and more coherent; however, in cases like the fourth example, the model is not able to perform logical reasoning (negation). Moreover, the model fails in cases where the input entity is not recognized correctly as in case of the last example. Although, the name of the friend is fetched from the vocabulary, but the predicted entity is wrong hence the model is not able to decode the relation $r$:address from the KG during decoding.\n\\vspace{-.8cm}\n\\begin{table}[ht]\n\n \\caption{Error analysis of KGIRNet.}\n\n\\begin{adjustbox}{width=\\textwidth}\n\\begin{tabular} {p{42mm}|p{50mm}|p{50mm}}\n\n\\toprule\n{\\textbf{Input Query}} &  \\textbf{True Response} & \n\\textbf{Predicted Response}\\\\\n\\midrule\n\n{portugal win the last eurocup ?} & {yes they won the last europcup} & \\textit{portugal has won the world cup.} \\\\\n\\midrule\n{who is switzerland\\'s best player ?} & {xherdan shaqiri} & \\textit{petkovic is the coach of switzerland } \\\\ \n\\midrule\n{who is considered their best player ?} & {traditionally their top scorers.} & \\textit{luis suarez is the best player in argentina} \\\\ \n\\midrule\n{will it be warm in grand\\_rapids on saturday } & {it will not be warm in grand\\_rapids on saturday} & \\textit{it is not predicted to fall on tuesday in grand\\_rapids} \\\\ \n\\midrule\n{where does my friend live ?} & {tom lives at 580\\_van\\_ness\\_ave} & \\textit{tom s house is located at $r$:address} \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n       \\label{tab:errors}\n\\end{table}\n\\vspace{-.6cm}\nIn general, the model's entity detection fails in case of conversations with long dialogue contexts. Incorporating KG information into the entity detection process, where we consider the sub-graph of the entity in the first interaction can potentially help in further performance improvements.\n\n\\section{Related Work}\nLeveraging background information for dialogue system improvement is a well-researched topic, especially in goal-oriented setting~\\cite{bordes2016learning,dhingra2016towards,wen2016network}. \\cite{eric2017key} proposed the in-car dataset which uses a knowledge base for in-car conversation about weather, location etc. Recently, \\cite{madotto2018mem2seq} proposed memory-network based encoder-decoder architecture for integrating knowledge into the dialogue generation process on this dataset. Improved models in this task are proposed by\n\\cite{kassawat2019incorporating,wu2019global}. \n\\cite{chaudhuri2019using} proposed a soccer dialogue dataset along with a KG-Copy mechanism for non-goal oriented dialogues which are KG-integrated. In a slightly similar research line, in past years, we also notice the use of variational autoencoders (VAE)~\\cite{zhao-etal-2017-learning,li-etal-2020-improving-variational} and generative adversarial networks (GANs)~\\cite{olabiyi-etal-2019-multi,lopez2019differentiable} in dialogue generation. However, knowledge graph based dialogue generation is not well-explored in these approaches.\n\nMore recently, transformer-based~\\cite{vaswani2017attention} pre-trained models have achieved success in solving various downstream tasks in the field of NLP such as question answering~\\cite{shao2019transformer} \\cite{devlin2019bert}, machine translation~\\cite{wang2019learning}, summarization~\\cite{egonmwan2019transformer}. Following the trend, a hierarchical transformer is proposed by~\\cite{santra2020hierarchical} for task-specific dialogues. The authors experimented on MultiWOZ dataset~\\cite{budzianowski2018multiwoz}, where the belief states are not available. However, they found the use of hierarchy based transformer models effective in capturing the context and dependencies in task-specific dialogue settings. In a different work,~\\cite{oluwatobi2020dlgnet} experimented transformer-based model on both the task-specific and non-task specific dialogues in multi-turn setting. In a recent work, ~\\cite{zhang2019dialogpt} investigated on transformer-based model for non-goal oriented dialogue generation in single-turn dialogue setting. Observing the success of transformer-based models over the recurrent models in this paper we also employ BERT in the dialogue generation process which improves the quality of generated dialogues (discussed in section~\\ref{results} and~\\ref{discussion}).\n\nIn past years, there is a lot of focus on encoding graph structure using neural networks, a.k.a.~Graph Neural Networks (GNNs)~\\cite{bronstein2017geometric,velivckovic2017graph}. In the field of computer vision, Convolutional Neural Networks (CNNs) are used to extract the most meaningful information from grid-like data structures such as images. A generalization of CNN to graph domain, Graph Convolutional Networks (GCNs)~\\cite{kipf2016semi} has become popular in the past years. Such architectures are also adapted for encoding and extracting information from knowledge graphs \\cite{neil2018interpretable}. Following a similar research line, in this paper, we leverage the concept of Graph Laplacian~\\cite{kipf2016semi} for encoding sub-graph information into the learning mechanism.\n\n\\section{Conclusion and Future Work} \nIn this paper, we have studied the task of generating knowledge grounded dialogues. We bridged the gap between two well-researched topics, namely knowledge grounded question answering and end-to-end dialogue generation. We propose a novel decoding method which leverages pre-trained transformers, KG structure and Graph Laplacian based encoding during the response generation process. Our evaluation shows that out proposed model produces better knowledge grounded response compared to other state-of-the-art approaches, for both the task and non-task oriented dialogues. \n\nAs future work, we would like to focus on models with better understanding of text in order to perform better KG based reasoning. We also aim to incorporate additional KG structure information in the entity detection method. Further, a better handling of infrequent relations seen during training may be beneficial.\n\n\\section*{Acknowledgement}\nWe acknowledge the support of the excellence clusters ScaDS.AI (BmBF IS18026A-F), ML2R (BmBF FKZ 01 15 18038 A\/B\/C), TAILOR (EU GA 952215) and the  projects SPEAKER (BMWi FKZ 01MK20011A) and JOSEPH (Fraunhofer Zukunftsstiftung).\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}