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）
Jiunn-Wei Chen, Yu-Ting Chen, Ghanashyam Meher, Berndt Müller, Andreas Schäfer, Xiaojun Yao
We simulate the local thermalization dynamics for minimally truncated SU(2) pure gauge theory on linear plaquette chains with up to 151 plaquettes using IBM quantum computers. We study the time dependence of the entanglement spectrum, Rényi-2 entropy and anti-flatness on small subsystems. The quantum hardware results obtained after error mitigation agree with extrapolated classical simulator results for chains consisting of up to 101 plaquettes. Our results demonstrate the feasibility of local thermalization studies for chaotic quantum systems, such as nonabelian lattice gauge theories, on current noisy quantum computing platforms.
In this talk, we discuss real-time thermalization dynamics of $\mathbf{Z}_2$ Lattice Gauge Theory in 2+1 spacetime dimensions. While classical thermalization is commonly associated with chaotic behavior, turbulence and universality, the manifestation of these phenomena in quantum mechanical systems is not clear. However, when viewed through the lens of Entanglement Structure, we find that quantum thermalization proceeds in characteristic stages and reveals phenomena remarkably similar to their classical counterparts: chaos, turbulence and universality.
Using dual theories embedded into a larger unphysical Hilbert space along entanglement cuts, we study the Entanglement Structure of $\mathbf{Z}_2$ lattice gauge theory in $(2+1)$ spacetime dimensions. We demonstrate Li and Haldane's conjecture, and show consistency of the Entanglement Hamiltonian with the Bisognano-Wichmann theorem. Studying non-equilibrium dynamics after a quench, we provide an extensive description of thermalization in $\mathbf{Z}_2$ gauge theory which proceeds in a characteristic sequence: Maximization of the Schmidt rank and spreading of level repulsion at early times, sel…
We determine the time evolution of fluctuations of the Polyakov loop after a quench into the deconfined phase of SU(3) gauge theory from a simple classical relativistic Lagrangian. We compare the structure factors, which indicate spinodal decomposition followed by relaxation, to those obtained via Markov Chain Monte Carlo techniques in SU(3) lattice gauge theory. We find that the time when the structure factor peaks diverges like $\sim 1/k^2$ in the long-wavelength limit. This is due to formation of competing Z(3) domains for configurations where the Polyakov loop exhibits non-perturbatively l…
Thermalization of a heavy quark near rest is controlled by the correlator of two electric fields along a temporal Wilson line. We address this correlator within real-time, classical lattice Yang-Mills theory, and elaborate on the analogies that exist with the dynamics of hot QCD. In the weak-coupling limit, it can be shown analytically that the dynamics on the two sides are closely related to each other. For intermediate couplings, we carry out non-perturbative simulations within the classical theory, showing that the leading term in the weak-coupling expansion significantly underestimates the…
I propose a method, based on a set of Langevin equations, for bringing classical gauge theories to thermal equilibrium while respecting the set of Gauss' constraints exactly. The algorithm is described in detail for the SU(2) gauge theory with or without the Higgs doublet. As an example of application, canonical average of the maximal Lyapunov exponent is computed for the SU(2) Yang-Mills theory.
Gauge theories form the foundation of modern physics, with applications ranging from elementary particle physics and early-universe cosmology to condensed matter systems. We perform quantum simulations of the unitary dynamics of a U(1) symmetric gauge field theory and demonstrate emergent irreversible behavior. The highly constrained gauge theory dynamics is encoded in a one-dimensional Bose--Hubbard simulator, which couples fermionic matter fields through dynamical gauge fields. We investigate global quantum quenches and the equilibration to a steady state well approximated by a thermal ensem…
Non-Abelian string melting and thermalization in an open lattice gauge theory
Ludwig Maximilian University of Munich（慕尼黑大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；Dipartimento di Fisica, Università di Bari（巴里大学物理系）；INFN, Sezione di Bari（意大利国家核物理研究所巴里分部）；Department of Physics, College of Science, Kyung Hee University（庆熙大学理学院物理系）
We investigate thermalization processes occurring at different time scales in the Yang-Mills-Higgs system at high temperatures. We determine the largest Lyapunov exponent associated with the gauge fields and verify its relation to the perturbatively calculated damping rate of a static gauge boson.
We analyze the Hamiltonian time evolution of classical SU(2) Yang-Mills-Higgs theory with a fundamental Higgs doublet on a spacial lattice. In particular, we study energy transfer and equilibration processes among the gauge and Higgs sectors, calculate the maximal Lyapunov exponents under randomized initial conditions in the weak-coupling regime, where one expects them to be related to the high-temperature plasmon damping rate, and investigate their energy and coupling dependence. We further examine finite-time and finite-size errors, study the impact of the Higgs fields on the instability of …
We perform SU(3) Lattice Gauge Theory simulations of the deconfinement transition attempting to mimic conditions encountered in heavy ion collisions. Specifically, we perform a sudden temperature quench across the deconfinement temperature, and follow the response of the system in successive simulation sweeps under spatial lattice expansion and temperature fall-off. In measurements of the Polyakov loop and structure functions a robust strong signal of global instability response is observed through the exponential growth of low momentum modes. Development of these long range modes isotropizes …
We present simulations of non-equilibrium dynamics of quantum field theories on digital quantum computers. As a representative example, we consider the Schwinger model, a 1+1 dimensional U(1) gauge theory, coupled through a Yukawa-type interaction to a thermal environment described by a scalar field theory. We use the Hamiltonian formulation of the Schwinger model discretized on a spatial lattice. With the thermal scalar fields traced out, the Schwinger model can be treated as an open quantum system and its real-time dynamics are governed by a Lindblad equation in the Markovian limit. The inte…
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.
As a toy model for dynamics in nonequilibrium quantum field theory we consider the abelian Higgs model in 1+1 dimensions with fermions. In the approximate dynamical equations, inhomogeneous classical (mean) Bose fields are coupled to quantized fermion fields, which are treated with a mode function expansion. The effective equations of motion imply e.g. Coulomb scattering, due to the inhomogeneous gauge field. The equations are solved numerically. We define time dependent fermion particle numbers with the help of the single-time Wigner function and study particle production starting from inhomo…
We excite the vacuum of a relativistic theory of bosons coupled to a $U(1)$ gauge field in 1+1 dimensions (bosonic Schwinger model) out of equilibrium by creating a spatially separated particle-antiparticle pair connected by a string of electric field. During the evolution, we observe a strong confinement of bosons witnessed by the bending of their light cone, reminiscent of what has been observed for the Ising model [Nat. Phys. 13, 246 (2017)]. As a consequence, for the time scales we are able to simulate, the system evades thermalization and generates exotic asymptotic states. These states a…
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…
Local gauge symmetry is intriguing for the study of quantum thermalization breaking. For example, in the high-spin lattice Schwinger model (LSM), the local U(1) gauge symmetry underlies the disorder-free many-body localization (MBL) dynamics of matter fields. This mechanism, however, would not work in a spin-1/2 LSM due to the absence of electric energy in the Hamiltonian. In this paper, we show that the spin-1/2 LSM can also exhibit disorder-free MBL dynamics, as well as entropy prethermalization, by introducing a four-fermion interaction into the system. The interplay between the fermion int…
Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）；Department of Physics, Duke University（杜克大学物理系）；Institut für Theoretische Physik, Universität Regensburg（雷根斯堡大学理论物理研究所）；Department of Physics, National Taiwan University（台湾大学物理系）；Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig Maximilian University of Munich（慕尼黑大学物理系与阿诺德·索末菲理论物理中心）；InQubator for Quantum Simulation, Department of Physics, University of Washington（华盛顿大学物理系量子模拟创新中心）
A heatbath algorithm is proposed for pure SU(N) lattice gauge theory based on the Manton action of the plaquette element for general gauge group N. Comparison is made to the Metropolis thermalization algorithm using both the Wilson and Manton actions. The heatbath algorithm is found to outperform the Metropolis algorithm in both execution speed and decorrelation rate. Results, mostly in D=3, for N=2 through 5 at several values for the inverse coupling are presented.
An algorithm is proposed for the simulation of pure SU(N) lattice gauge theories based on Genetic Algorithms(GAs). We apply GAs to SU(2) pure gauge theory on a 2 dimensional lattice and show the results, the action per plaquette and Wilson loops, are consistent with those by Metropolis method(MP)s and Heatbath method(HB)s. Thermalization speed of GAs is especially faster than the simple MPs.
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.
We review recent results from studies of the dynamics of classical Yang-Mills fields on a lattice. We discuss the numerical techniques employed in solving the classical lattice Yang-Mills equations in real time, and present results exhibiting the universal chaotic behavior of nonabelian gauge theories. The complete spectrum of Lyapunov exponents is determined for the gauge group SU(2). We survey results obtained for the SU(3) gauge theory and other nonlinear field theories. We also discuss the relevance of these results to the problem of thermalization in gauge theories.
We investigate the effects of dissipation in the deconfinement transition for pure SU(2) and SU(3) gauge theories. Using an effective theory for the order parameter, we study its Langevin evolution numerically. Noise effects are included for the case of SU(2). We find that both dissipation and noise have dramatic effects on the spinodal decomposition of the order parameter and delay considerably its thermalization. For SU(3) the effects of dissipation are even larger than for SU(2).
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…
We test at the electroweak scale the recently proposed elaborate theoretical scenario for real-time dynamics of non-abelian gauge theories at high temperature. We see no sign of the predicted behavior. This indicates that perturbative concepts like color conductivity and Landau damping might be irrelevant at temperatures corresponding to the electroweak scale.
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…
An Algorithm is proposed for the simulation of pure SU(N) lattice gauge theories based on Genetic Algorithms(GAs). Main difference between GAs and Metropolis methods(MPs) is that GAs treat a population of points at once, while MPs treat only one point in the searching space. This provides GAs with information about the assortment as well as the fitness of the evolution function and producting a better solution. We apply GAs to SU(2) pure gauge theory on a 2 dimensional lattice and show the results are consistent with those given by MP and Heatbath methods(HBs). Thermalization speed of GAs is e…
We derive a representation for a lattice U(1) gauge theory with exponential convergence in the number of states used to represent each lattice site that is applicable at all values of the coupling. At large coupling, this representation is equivalent to the Kogut-Susskind electric representation, which is known to provide a good description in this region. At small coupling, our approach adjusts the maximum magnetic field that is represented in the digitization as in this regime the low-lying eigenstates become strongly peaked around zero magnetic field. Additionally, we choose a representatio…
Preparing strongly-coupled particle states on quantum computers requires large resources. In this work, we show how classical sampling coupled with projection operators can be used to compute Minkowski matrix elements without explicitly preparing these states on the quantum computer. We demonstrate this for the 2+1d $\mathbb{Z}_2$ lattice gauge theory on small lattices with a quantum simulator.
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…
It is widely anticipated that a large-scale quantum computer will offer an evermore accurate simulation of nature, opening the floodgates for exciting scientific breakthroughs and technological innovations. Here, we show a complete, instruction-by-instruction rubric to simulate U(1), SU(2), and SU(3) lattice gauge theories on a quantum computer. These theories describe quantum electrodynamics and chromodynamics, the key ingredients that form the fabric of our universe. We further provide a concrete estimate of the quantum computational resources required for an accurate simulation of lattice g…
We propose a new local algorithm for the thermalization of n-vector spin models, which can also be used in the numerical simulation of SU(N) lattice gauge theories. The algorithm combines heat-bath (HB) and micro-canonical updates in a single step -- as opposed to the hybrid overrelaxation method, which alternates between the two kinds of update steps -- while preserving ergodicity. We test our proposed algorithm in the case of the one-dimensional 4-vector spin model and compare its performance with the standard HB algorithm and with other HB-inspired algorithms.
Efficient digitization is required for quantum simulations of gauge theories. Schemes based on discrete subgroups use fewer qubits at the cost of systematic errors. We systematize this approach by deriving a single plaquette action for approximating general continuous gauge groups through integrating out field fluctuations. This provides insight into the effectiveness of these approximations, and how they could be improved. We accompany the scheme by simulations of pure gauge over the largest discrete subgroup of $SU(3)$ up to the third order.
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.
The discovery of the fact that black holes radiate particles and eventually evaporate led Hawking to pose the well-known information loss paradox. This paradox caused a long and serious debate since it claims that the fundamental laws of quantum mechanics may be violated. A possible cure appeared recently from superstring theory, a consistent theory of quantum gravity: if the holographic description of a quantum black hole based on the gauge/gravity duality is correct, the information is not lost and quantum mechanics remains valid. Here we test this gauge/gravity duality on a computer at the …
We present a multiscale thermalization algorithm for lattice gauge theory, which enables efficient parallel generation of uncorrelated gauge field configurations. The algorithm combines standard Monte Carlo techniques with ideas drawn from real space renormalization group and multigrid methods. We demonstrate the viability of the algorithm for pure Yang-Mills gauge theory for both heat bath and hybrid Monte Carlo evolution, and show that it ameliorates the problem of topological freezing up to controllable lattice spacing artifacts.
The simulation of real-time dynamics in lattice gauge theories is particularly hard for classical computing due to the exponential scaling of the required resources. On the other hand, quantum algorithms can potentially perform the same calculation with a polynomial dependence on the number of degrees of freedom. A precise estimation is however particularly challenging for the simulation of lattice gauge theories in arbitrary dimensions, where, gauge fields are dynamical variables, in addition to the particle fields. Moreover, there exist several choices for discretizing particles and gauge fi…
In the near-future noisy intermediate-scale quantum (NISQ) era of quantum computing technology, applications of quantum computing will be limited to calculations of very modest scales in terms of the number of qubits used. The need to represent numeric quantities using limited resources leads to digitization errors which must be taken into account. As a first step towards quantum simulations of realistic high-energy physics problems, we explore classically the effects of digitizing elements of the $\mathrm{SU}(2)$ gauge group to a finite set. We consider several methods for digitizing the grou…
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 compute nonequilibrium dynamics for classical-statistical SU(2) pure gauge theory on a lattice. We consider anisotropic initial conditions with high occupation numbers in the transverse plane on a characteristic scale ~ Q_s. This is used to investigate the very early stages of the thermalization process in the context of heavy-ion collisions. We find Weibel or "primary" instabilities with growth rates similar to those obtained from previous treatments employing anisotropic distributions of hard modes (particles) in the weak coupling limit. We observe "secondary" growth rates for higher-mome…
