The quantum link~\cite{Brower:1997ha} Hamiltonian was introduced two decades ago as an alternative to Wilson's Euclidean lattice QCD with gauge fields represented by bi-linear fermion/anti-fermion operators. When generalized this new microscopic representation of lattice field theories is referred as {\tt D-theory}~\cite{Brower:2003vy}. Recast as a Hamiltonian in Minkowski space for real time evolution, D-theory leads naturally to quantum Qubit algorithms. Here to explore digital quantum computing for gauge theories, the simplest example of U(1) compact QED on triangular lattice is defined and gauge invariant kernels for the Suzuki-Trotter expansions are expressed as Qubit circuits capable of being tested on the IBM-Q and other existing Noisy Intermediate Scale Quantum (NISQ) hardware. This is a modest step in exploring the quantum complexity of D-theory to guide future applications to high energy physics and condensed matter quantum field theories.
The Hamiltonian approach can be used successfully to study the real-time evolution of a non-Abelian lattice gauge theory on the available noisy quantum computers. In this work, results from the real-time evolution of SU(2) pure gauge theory on IBM hardware are presented. The long real-time evolution spanning dozens of Trotter steps with hundreds of CNOT gates and the observation of a traveling excitation on the lattice were made possible by using a collection of error mitigation techniques. Self-mitigation is our novel tool, which consists of using the same physics circuit as a noise-mitigatio…
Quantum computers offer the possibility to implement lattice gauge theory in Minkowski rather than Euclidean spacetime, thus allowing calculations of processes that evolve in real time. In this work, calculations within SU(2) pure gauge theory are able to show the motion of an excitation traveling across a spatial lattice in real time. This is accomplished by using a simple yet powerful method for error mitigation, where the original circuit is used both forward and backward in time. For a two-plaquette lattice, meaningful results are obtained from a circuit containing hundreds of CNOT gates. …
We realize, for the first time, a non-Abelian gauge theory with both gauge and matter fields on a quantum computer. This enables the observation of hadrons and the calculation of their associated masses. The SU(2) gauge group considered here represents an important first step towards ultimately studying quantum chromodynamics, the theory that describes the properties of protons, neutrons and other hadrons. Quantum computers are able to create important new opportunities for ongoing essential research on gauge theories by providing simulations that are unattainable on classical computers. Our c…
Although the Hamiltonian formalism is so far favored for quantum computation of lattice gauge theory, the path integral formalism would never be useless. The advantages of the path integral formalism are the knowledge and experience accumulated by classical lattice simulation and manifest Lorentz invariance. We discuss quantum computation of lattice gauge theory in the path integral formalism. We utilize a quantum sampling algorithm to generate gauge configurations, and demonstrate a benchmark test of $Z_2$ lattice gauge theory on a four-dimensional hypercube.
We propose a simple method to construct staples in lattice gauge theory with Wilson action on a parallel computer. This method can be applicable to any dimensional system and to any dimensional division without difficulty. Furthermore this requires rather small working area to realize gauge simulation on a parallel computer.
Relativistic spin 1/2, as represented by Susskind's 1977 discretization of the Dirac equation on a spatial lattice, is shown to follow from basic, not typically relativistic but essentially quantum theoretic assumptions: that position eigenstates propagate to nearest neighbours while respecting lattice symmetries modulo gauge transformations.
Aiming at evading the notorious sign problem in classical Monte-Carlo approaches to lattice quantum chromodynamics, we present an approach for quantum computing finite-temperature lattice gauge theories at non-zero density. Based on the thermal pure-quantum-state formalism of statistical mechanics when extended to gauge-theory systems, our approach allows for sign-problem-free quantum computations of thermal expectation values and non-equal time correlation functions. By taking a simple lattice gauge theory for which classical benchmarks are possible, namely $\mathbb{Z}_2$ lattice gauge theory…
The simulation of dense fermionic matters is a long-standing problem in lattice gauge theory. One hopeful solution would be the use of quantum computers. In this paper, digital quantum simulation is designed for lattice gauge theory at nonzero density. The quantum variational algorithm is adopted to obtain the ground state at nonzero density. A benchmark test is performed in the lattice Schwinger model.
We propose a method which allows the generalization of the Landau lattice gauge-fixing procedure to generic covariant gauges. We report preliminary numerical results showing how the procedure works for $SU(2)$ and $SU(3)$. We also report numerical results showing that the contribution of finite lattice-spacing effects and/or spurious copies are relevant in the lattice gauge-fixing procedure.
Gauged gaussian Projected Entangled Pair States are particular tensor network constructions that describe lattice states of fermionic matter interacting with dynamical gauge fields. We show how one can efficiently compute, using Monte-Carlo techniques, expectation values of physical observables in that class of states. This opens up the possibility of using tensor network techniques to investigate lattice gauge theories in two and three spatial dimensions.
Dynamical gauge fields are essential to capture the short and large distance behavior of gauge theories (confinement, mass gap, chiral symmetry breaking, asymptotic freedom). I propose two possible strategies to use optical lattices to mimic simulations performed in lattice gauge theory. I discuss how new developments in optical lattices could be used to generate local invariance and link composite operators with adjoint quantum numbers that could play a role similar to the link variables used in lattice gauge theory. This is a slightly expanded version of a poster presented at the KITP Confer…
We investigate the dynamics of lattice gauge theories in an Abelian monopole background field. By means of the gauge-invariant lattice Schrodinger functional we study the Abelian monopole condensation in U(1) lattice gauge theory at zero temperature and in SU(3) lattice gauge theory at finite temperature.
An improved mapping of one-dimensional SU(2) non-Abelian gauge theory onto qubit degrees of freedom is presented. This new mapping allows for a reduced unphysical Hilbert space. Insensitivity to interactions within this unphysical space is exploited to design more efficient quantum circuits. Local gauge symmetry is used to analytically incorporate the angular momentum alignment, leading to qubit registers encoding the total angular momentum on each link. The results of a multi-plaquette calculation on IBM's quantum hardware are presented.
The scaling slope of the anti-symmetric mass gap M of compact U(1)_{2+1} lattice gauge theory is obtained analytically in the Hamiltonian formalism using the plaquette expansion. Based on the first four moments of the Hamiltonian with respect to a one-plaquette mean field state the results demonstrate clear scaling of M at and beyond the transition from strong to weak coupling. The scaling parameters determined agree well with the range of numerical determinations available.
We address the problem of the gauge fixing versus Gribov copies in lattice gauge theories. For the Landau gauge, results show that a suitable combination of evolutionary algorithms with traditional steepest descent methods identifies the global maximum of the optimisation function. We discuss the performance of the combined algorithm on small cubic lattices for SU(2) and SU(3).
We propose a non-perturbative procedure to fix generic covariant gauges on the lattice. Varying the gauge parameter, this gauge fixing provides a concrete method to check numerically the gauge dependence of correlators measured on the lattice. The new algorithm turns out to converge with a good efficiency. As a preliminary physical result, we find a sensitive dependence of the gluon propagator on the gauge parameter.
The architecture and capabilities of the computers currently in use for large-scale lattice QCD calculations are described and compared. Based on this present experience, possible future directions are discussed.
After an introduction in which we review the fundamental difficulty in constructing lattice chiral gauge theories, we summarize the analytic and numerical evidence that abelian lattice chiral gauge theories can be nonperturbatively constructed through the gauge-fixing approach. In addition, we indicate how we believe that the method may be extended to nonabelian chiral gauge theories.
We study Abelian monopole and vortex condensation in lattice pure gauge theories. Condensation is detected by means of a disorder parameter defined in terms of a gauge-invariant effective action introduced using the lattice Schrödinger functional. Dirac monopoles condense in the confined phase of U(1) lattice gauge theory. Abelian monopoles and Abelian vortices condense in the confined phase of SU(2) and SU(3) lattice gauge theories.
We introduce a generalized Maximum Abelian Gauge (MAG). We work with this new gauge on 12^4 lattices for beta=5.7,5.8 and 16^4 lattices for beta=5.9,6.0. We also introduce a form of abelian projection related to the generalized MAG. We measure U(1)xU(1) wilson loops and single color magnetic current densities.
We define a benchmark suite for lattice QCD and report on benchmark results from several computer platforms. The platforms considered are apeNEXT, CRAY T3E, Hitachi SR8000, IBM p690, PC-Clusters, and QCDOC.
A new gauge fixing condition is discussed, which is (lattice) rotation invariant, has the `smoothness' properties of the Landau gauge but can be efficiently computed and is unambiguous for almost all lattice gauge field configurations.
Lattice gluon propagators are studied using tadpole and Symanzik improved gauge action in Landau gauge. The study is performed using anisotropic lattices with asymmetric volumes. The Landau gauge dressing function for the gluon propagator measured on the lattice is fitted according to a leading power behavior: $Z(q^2)\simeq (q^2)^{2κ}$ with an exponent $κ$ at small momenta. The gluon propagators are also fitted using other models and the results are compared. Our result is compatible with a finite gluon propagator at zero momentum in Landau gauge.
On the lattice some of the salient features of pure gauge theories and of gauge theories with fermions in complex representations of the gauge group seem to be lost. These features can be recovered by considering part of the theory in the continuum. The prerequisite for that is the construction of continuum gauge fields from lattice gauge fields. Such a construction, which is gauge covariant and complies with geometrical constructions of the topological charge on the lattice, is given in this paper. The procedure is explicitly carried out in the $U(1)$ theory in two dimensions, where it leads …
After an introduction in which we review the fundamental difficulty in constructing lattice chiral gauge theories, we discuss the analytic and numerical evidence that abelian lattice chiral gauge theories can be non-perturbatively constructed through the gauge-fixing approach. While a complete non-abelian extension is still under construction, we also show how fermion-number violating processes are realized in this approach.
The importance of lattice gauge field interpolation for our recent non-perturbative formulation of chiral gauge theory is emphasized. We illustrate how the requisite properties are satisfied by our recent four-dimensional non-abelian interpolation scheme, by going through the simpler case of U(1) gauge fields in two dimensions.
Parallel computers dedicated to lattice field theories are reviewed with emphasis on the three recent projects, the Teraflops project in the US, the CP-PACS project in Japan and the 0.5-Teraflops project in the US. Some new commercial parallel computers are also discussed. Recent development of semiconductor technologies is briefly surveyed in relation to possible approaches toward Teraflops computers.
We develop a method for calculating the Lyapunov characteristic exponents of lattice gauge theories. The complete Lyapunov spectrum of SU(2) gauge theory is obtained and Kolmogorov-Sinai entropy is calculated. Rapid convergence with lattice size is found.
We solve the Gauss law of SU(2) lattice gauge theory using the harmonic oscillator prepotential formulation. We construct a generating function of a manifestly gauge invariant and orthonormal basis in the physical Hilbert space of (d+1) dimensional SU(2) lattice gauge theory. The resulting orthonormal physical states are given in closed form. The generalization to SU(N) gauge group is discussed.
A gauge invariant procedure for extracting combined SO(3)-Z2 monopoles in positive-plaquette SU(2) lattice gauge theory is shown. When these monopoles are eliminated through a constraint, the theory deconfines for all $β$ on $12^4$ and $20^4$ lattices even in the strong coupling limit, despite a rather strong average plaquette of around 0.64. This corresponds to an effective Wilson $β$ of 2.45, at which Wilson-action lattices would be far into the confining region. This suggests that Wilson-action confinement may be a strong-coupling lattice artifact; the continuum limit may not confine.
I review our method to formulate chiral gauge theories on the lattice based on a two-cutoff lattice regularization, and discuss recent numerical results in a chiral U(1) gauge theory in 2D.
We address the problem of defining the gauge four-potential on the lattice, in terms of the natural link variables. Different regularized definitions are shown, through non perturbative numerical computation, to converge towards the same continuum renormalized limit.
The use of lattice tensor representations is explored to investigate the lattice Landau gauge gluon propagator for the pure SU(3) Yang-Mills gauge theory in 4D. The analysis of several tensor bases allows to quantify the completeness of the various tensor bases considered, the deviations of the lattice results from the continuum theory due to the lattice artefacts and estimate the theoretical uncertainty in the propagator. Furthermore, our analysis tests continuum based relations with the lattice data and show that the lattice Landau gauge gluon propagator is described by a unique form factor,…
The lattice formulation of gauge theories in a renormalizable gauge is discussed. The formulation invokes a new phase diagram, and it may allow for a lattice definition of Chiral Gauge Theories.
Local Thermalization of SU(2) Lattice Gauge Fields on Quantum Computers
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；University of Washington（华盛顿大学）；Duke University（杜克大学）；Universität Regensburg（雷根斯堡大学）
We identify the artifact that causes confinement at strong coupling in compact U(1) lattice gauge theory and show that most of the string tension in compact SU(2) lattice gauge theory is due to plaquettes of negative trace.
We discuss two proposals for a non-perturbative formulation of chiral gauge theories on the lattice. In both cases gauge symmetry is broken by the regularization. We aim at a dynamical restoration of symmetry. If the gauge symmetry breaking is not too severe this procedure could lead in the continuum limit to the desired chiral gauge theory.
We report on recent progress with the gauge-fixing approach to lattice chiral gauge theories. The bosonic sector of the gauge-fixing approach is studied with fully dynamical U(1) gauge fields. We demonstrate that it is important to formulate the Lorentz gauge-fixing action such that the dense set of lattice Gribov copies is removed, and the gauge-fixing action has a unique absolute minimum. We then show that the spectrum in the continuum limit contains only the desired massless photon, as expected.
We address the problem of defining the gluon field on the lattice in terms of the natural link variables. Different regularized definitions are shown, through non perturbative numerical computation, to converge towards the same continuum renormalized limit.
We present results for the fundamental string tension in SU(2) lattice gauge theory after projection to maximal abelian and direct maximal center gauges. We generate 20 Gribov copies/configuration. Abelian and center projected string tensions slowly decrease as higher values of the gauge functionals are reached.
The new method of nonperturbative calculation of the beta function in the lattice gauge theory is proposed. The method is based on the finite size scaling hypothesis.
The 4-d SU(2) lattice gauge theory is simulated in the minimal Coulomb gauge which aims to maximize the traces of all links in three directions. Fourth-direction links are interpreted as spins in a Heisenberg-like model with varying interactions. These spins magnetize in 3-d hyperlayers at weak coupling, breaking a remnant gauge symmetry, as well as the Polyakov-loop symmetry. They demagnetize at a phase transition around $β= 2.5$ on the infinite lattice, as determined by Binder cumulant crossings. Because $N$ symmetries are breaking on an $N^4$ lattice, the transition is unusually broad, enco…
The partition function (quantum transition amplitude) of the gauge system with gauge group $Z_2$ coupled with Majorana fermions is calculated on the regular 3D cubic lattice.
The traditional approach for studying the physics of the strong interactions employs a basic computational construct originally proposed by Wilson in the 1970s. Over the years additional enhancements have been added to this formulation to improve computational performance and accuracy. This formulation has been successfully implemented on high performance computing systems and has yielded accurate calculations for many static properties of the strong interactions (such as the hadron mass spectrum). With the recent advances in quantum computing, the question that is now being asked is whether a…
We present a scheme for the analytic computation of renormalization functions on the lattice, using a symbolic manipulation computer language. Our first nontrivial application is a new three-loop result for the topological susceptibility.
We demonstrate that a direct approach to improving Hamiltonian lattice gauge theory is possible. Our approach is to correct errors in the Kogut-Susskind Hamiltonian by incorporating additional gauge invariant terms. The coefficients of these terms are chosen so that the order $a^2$ classical errors vanish. We conclude with a brief discussion of tadpole improvement in Hamiltonian lattice gauge theory.
We illustrate preliminary results on the gluon propagator computed in generic covariant lattice gauges in the quenched approximation with the Wilson action. We have applied a recently proposed procedure to fix a generic covariant gauge on the lattice. We make a comparison among results obtained at two different values of the gauge parameter and in the standard Landau's gauge.
This is a review, intended for lattice nonspecialists, of the studies of the compact abelian gauge theories on the lattice performed by the Aachen lattice field theory group. We discuss in particular the pure compact QED and a U(1) lattice gauge theory with charged scalar and fermion fields in four and three dimensions. Our data indicate that these lattice theories might define some continuum field theories, or at least effective field theories with remarkable nonperturbative properties like confinement and dynamical mass generation.
Applying Genetic Algorithm for the Lattice Gauge Theory is formed to be an effective method to minimize the action of gauge field on a lattice. In 4 dimensions, the critical point and the Wilson loop behaviour of SU(2) lattice gauge theory as well as the phase transition of U(1) theory have been studied. The proper coding methodi has been developed in order to avoid the increase of necessary memory and the overload of calculation for Genetic Algorithm. How hichhikers toward equlibrium appear against kidnappers is clarified.
It is described how to obtain an almost gauge invariant lattice action $S$ for chiral gauge theories, or other models in which a straightforward discretization leads to a lattice action $S^\pr$ in which gauge invariance is broken. The lattice action is `almost' gauge invariant, because a local gauge transformation leaves the action the same, up to a global gauge transformation. In this approach the action $S$ for all gauge fields on a gauge orbit is the same as that of the action $S^\pr$ evaluated for the gauge field fixed to a smooth gauge. To define $S$ unambiguously, it must be possible to …
