Lattice Gauge Theory Simulations Research Articles

Quantum error thresholds for gauge-redundant digitizations of lattice field theories

In the quantum simulation of lattice gauge theories, gauge symmetry can be either fixed or encoded as a redundancy of the Hilbert space. While gauge-fixing reduces the number of qubits, keeping the gauge redundancy can provide code space to mitigate and correct quantum errors by checking and restoring Gauss’s law. In this work, we consider the correctable errors for generic finite gauge groups and design the quantum circuits to detect and correct them. We calculate the error thresholds below which the gauge-redundant digitization with Gauss’s law error correction has better fidelity than the gauge-fixed digitization involving only gauge-invariant states. Our results provide guidance for fault-tolerant quantum simulations of lattice gauge theories. Published by the American Physical Society 2024

Nearly optimal state preparation for quantum simulations of lattice gauge theories

Nearly optimal state preparation for quantum simulations of lattice gauge theories

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Resource efficient schemes for the quantum simulation of lattice gauge theories can benefit from hybrid encodings of gauge and matter fields that use the native degrees of freedom, such as internal qubits and motional phonons in trapped-ion devices. We propose to use a parametric scheme to induce a tunneling of the phonons conditioned to the internal qubit state which, when implemented with a single trapped ion, corresponds to a minimal Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} gauge theory. To evaluate the feasibility of this scheme, we perform numerical simulations of the state-dependent tunneling using realistic parameters, and identify the leading sources of error in future experiments. We discuss how to generalize this minimal case to more complex settings by increasing the number of ions, moving from a single link to a Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} plaquette, and to an entire Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} chain. We present analytical expressions for the gauge-invariant dynamics and the corresponding confinement, which are benchmarked using matrix product state simulations.

Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements

Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements

Conserved energy–momentum tensor for real-time lattice simulations

We derive an expression for the energy–momentum tensor in the discrete lattice formulation of pure glue QCD. The resulting expression satisfies the continuity equation for energy conservation up to numerical errors with a symmetric procedure for the time discretization. In the case of the momentum conservation equation, we obtain an expression that is of higher accuracy in lattice spacing (O(a2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(a^2)$$\\end{document}) than the naive discretization where fields in the continuum expressions are replaced by discretized counterparts. The improvements are verified by performing numerical tests on the derived expressions using classical real-time lattice gauge theory simulations. We demonstrate substantial reductions in relative error of one to several orders of magnitude compared to a naive discretization for both energy and momentum conservation equations. We expect our formulation to have applications in the area of pre-equilibrium dynamics in ultrarelativistic heavy ion collisions, in particular for the extraction of transport coefficients such as shear viscosity.

Measurement-based quantum simulation of Abelian lattice gauge theories

Numerical simulation of lattice gauge theories is an indispensable tool in high energy physics, and their quantum simulation is expected to become a major application of quantum computers in the future. In this work, for an Abelian lattice gauge theory in dd spacetime dimensions, we define an entangled resource state (generalized cluster state) that reflects the spacetime structure of the gauge theory. We show that sequential single-qubit measurements with the bases adapted according to the former measurement outcomes induce a deterministic Hamiltonian quantum simulation of the gauge theory on the boundary. Our construction includes the (2+1)(2+1)-dimensional Abelian lattice gauge theory simulated on three-dimensional cluster state as an example, and generalizes to the simulation of Wegner’s lattice models M_{(d,n)}M(d,n) that involve higher-form Abelian gauge fields. We demonstrate that the generalized cluster state has a symmetry-protected topological order with respect to generalized global symmetries that are related to the symmetries of the simulated gauge theories on the boundary.Our procedure can be generalized to the simulation of Kitaev’s Majorana chain on a fermionic resource state. We also study the imaginary-time quantum simulation with two-qubit measurements and post-selections, and a classical-quantum correspondence, where the statistical partition function of the model M_{(d,n)}M(d,n) is written as the overlap between the product of two-qubit measurement bases and the wave function of the generalized cluster state.

Resource-efficient quantum simulation of lattice gauge theories in arbitrary dimensions: Solving for Gauss's law and fermion elimination

Resource-efficient quantum simulation of lattice gauge theories in arbitrary dimensions: Solving for Gauss's law and fermion elimination

Quantum error correction with gauge symmetries

Quantum simulations of lattice gauge theories (LGTs) are often formulated on an enlarged Hilbert space containing both physical and unphysical sectors in order to retain a local Hamiltonian. We provide simple fault-tolerant procedures that exploit such redundancy by combining a phase flip error correction code with the Gauss’ law constraint to correct one-qubit errors for a {{mathbb{Z}}}_{2} or truncated U(1) LGT in 1+1 and 2+1 dimensions with a link flux cutoff of 1. Unlike existing work on detecting violations of Gauss’ law, our circuits are fault tolerant and the overall error correction scheme outperforms a naïve application of the [5, 1, 3] code. The constructions outlined can be extended to LGT systems with larger cutoffs and may be of use in understanding how to hybridize error correction and quantum simulation for LGTs in higher space-time dimensions and with different symmetry groups.

Non-Abelian gauge invariance from dynamical decoupling

Lattice gauge theories are fundamental to such distinct fields as particle physics, condensed matter or quantum information theory. The recent progress in the control of artificial quantum systems already allows for studying Abelian lattice gauge theories in tabletop experiments. However, the realization of non-Abelian models remains challenging. Here, we employ a coherent quantum control scheme to enforce non-Abelian gauge invariance, and discuss this idea in detail for a one-dimensional SU(2) lattice gauge system. We comment on how to extend our scheme to other non-Abelian gauge symmetries and higher spatial dimensions. Because of its wide applicability, the presented coherent control scheme provides a promising route for the quantum simulation of non-Abelian lattice gauge theories.

Provably accurate simulation of gauge theories and bosonic systems

Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ−1, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ=polylog(ϵ−1).

Improved Hamiltonians for Quantum Simulations of Gauge Theories.

Quantum simulations of lattice gauge theories for the foreseeable future will be hampered by limited resources. The historical success of improved lattice actions in classical simulations strongly suggests that Hamiltonians with improved discretization errors will reduce quantum resources, i.e., require ≳2^{d} fewer qubits in quantum simulations for lattices with d-spatial dimensions. In this work, we consider O(a^{2})-improved Hamiltonians for pure gauge theories and design the corresponding quantum circuits for its real-time evolution in terms of primitive gates. An explicit demonstration for Z_{2} gauge theory is presented including exploratory tests using the ibm_perth device.

Digitising SU(2) gauge fields and the freezing transition

Efficient discretisations of gauge groups are crucial with the long term perspective of using tensor networks or quantum computers for lattice gauge theory simulations. For any Lie group other than U(1), however, there is no class of asymptotically dense discrete subgroups. Therefore, discretisations limited to subgroups are bound to lead to a freezing of Monte Carlo simulations at weak couplings, necessitating alternative partitionings without a group structure. In this work we provide a comprehensive analysis of this freezing for all discrete subgroups of SU(2) and different classes of asymptotically dense subsets. We find that an appropriate choice of the subset allows unfrozen simulations for arbitrary couplings, though one has to be careful with varying weights of unevenly distributed points. A generalised version of the Fibonacci spiral appears to be particularly efficient and close to optimal.

Preserving gauge invariance in neural networks

In these proceedings we present lattice gauge equivariant convolutional neural networks (L-CNNs) which are able to process data from lattice gauge theory simulations while exactly preserving gauge symmetry. We review aspects of the architecture and show how L-CNNs can represent a large class of gauge invariant and equivariant functions on the lattice. We compare the performance of L-CNNs and non-equivariant networks using a non-linear regression problem and demonstrate how gauge invariance is broken for non-equivariant models.

Quantum simulation of lattice gauge theories in more than one space dimension-requirements, challenges and methods.

Over recent years, the relatively young field of quantum simulation of lattice gauge theories, aiming at implementing simulators of gauge theories with quantum platforms, has gone through a rapid development process. Nowadays, it is not only of interest to the quantum information and technology communities. It is also seen as a valid tool for tackling hard, non-perturbative gauge theory problems by particle and nuclear physicists. Along the theoretical progress, nowadays more and more experiments implementing such simulators are being reported, manifesting beautiful results, but mostly on dimensional physics. In this article, we review the essential ingredients and requirements of lattice gauge theories in more dimensions and discuss their meanings, the challenges they pose and how they could be dealt with, potentially aiming at the next steps of this field towards simulating challenging physical problems in analogue, or analogue-digital ways.This article is part of the theme issue ‘Quantum technologies in particle physics’.

Photon-Mediated Stroboscopic Quantum Simulation of a Z2 Lattice Gauge Theory

Quantum simulation of lattice gauge theories (LGTs), aiming at tackling non-perturbative particle and condensed matter physics, has recently received a lot of interest and attention, resulting in many theoretical proposals, as well as several experimental implementations. One of the current challenges is to go beyond 1+1 dimensions, where four-body (plaquette) interactions, not contained naturally in quantum simulating devices, appear. In this Letter, we propose a method to obtain them based on a combination of stroboscopic optical atomic control and the non-local photon-mediated interactions appearing in nanophotonic or cavity QED setups. We illustrate the method for a $\mathbb{Z}_{2}$ lattice Gauge theory. We also show how to prepare the ground state and measure Wilson loops using state-of-the-art techniques in atomic physics.

Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories

Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for the purpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the most accurate, while computationally economic, Hamiltonian formulation(s) in such theories, considering the necessary truncation imposed on the Hilbert space of gauge bosons with any finite computing resources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs, which further require the imposition of non-Abelian Gauss's laws on the Hilbert space, introducing additional computational complexity. Focusing on the case of SU(2) LGT in 1+1 D coupled to matter, a number of different formulations of the original Kogut-Susskind framework are analyzed with regard to the dependence of the dimension of the physical Hilbert space on boundary conditions, system's size, and the cutoff on the excitations of gauge bosons. The impact of such dependencies on the accuracy of the spectrum and dynamics is examined, and the (classical) computational-resource requirements given these considerations are studied. Besides the well-known angular-momentum formulation of the theory, the cases of purely fermionic and purely bosonic formulations (with open boundary conditions), and the Loop-String-Hadron formulation are analyzed, along with a brief discussion of a Quantum Link Model of the same theory. Clear advantages are found in working with the Loop-String-Hadron framework which implements non-Abelian Gauss's laws a priori using a complete set of gauge-invariant operators. Although small lattices are studied in the numerical analysis of this work, and only the simplest algorithms are considered, a range of conclusions will be applicable to larger systems and potentially to higher dimensions.

ZN lattice gauge theory in a ladder geometry

Under the perspective of realizing analog quantum simulations of lattice gauge theories, ladder geometries offer an intriguing playground, relevant for ultracold atom experiments. Here, we investigate Hamiltonian lattice gauge theories defined in two-leg ladders. We consider a model that includes both gauge boson and Higgs matter degrees of freedom with local $\mathbb{Z}_N$ gauge symmetries. We study its phase diagram based on both an effective low-energy field theory and density matrix renormalization group simulations. For $N\ge 5$, an extended gapless Coulomb phase emerges, which is separated by a Berezinskii-Kosterlitz-Thouless phase transition from the surrounding gapped phase. Besides the traditional confined and Higgs regimes, we also observe a novel quadrupolar region, originated by the ladder geometry.

Quantum simulation of lattice gauge theories on superconducting circuits: Quantum phase transition and quench dynamics

Recently, quantum simulation of low-dimensional lattice gauge theories (LGTs) has attracted many interests, which may improve our understanding of strongly correlated quantum many-body systems. Here, we propose an implementation to approximate ℤ2 LGT on superconducting quantum circuits, where the effective theory is a mixture of a LGT and a gauge-broken term. By using matrix product state based methods, both the ground state properties and quench dynamics are systematically investigated. With an increase of the transverse (electric) field, the system displays a quantum phase transition from a disordered phase to a translational symmetry breaking phase. In the ordered phase, an approximate Gauss law of the ℤ2 LGT emerges in the ground state. Moreover, to shed light on the experiments, we also study the quench dynamics, where there is a dynamical signature of the spontaneous translational symmetry breaking. The spreading of the single particle of matter degree is diffusive under the weak transverse field, while it is ballistic with small velocity for the strong field. Furthermore, due to the emergent Gauss law under the strong transverse field, the matter degree can also exhibit confinement dynamics which leads to a strong suppression of the nearest-neighbor hopping. Our results pave the way for simulating the LGT on superconducting circuits, including the quantum phase transition and quench dynamics.

Toward scalable simulations of lattice gauge theories on quantum computers

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 fields on a lattice, each of them coming at different prices in terms of qubit register size and circuit depth. Here we provide a resource counting for real-time evolution of $U(1)$ gauge theories, such as Quantum Electrodynamics, on arbitrary dimension using the Wilson fermion representation for the particles, and the Quantum Link Model approach for the gauge fields. We study the phenomena of flux-string breaking up to a genuine bi-dimensional model using classical simulations of the quantum circuits, and discuss the advantages of our discretization choice in simulation of more challenging $SU(N)$ gauge theories such as Quantum Chromodynamics.

Lattice Gauge Theories and String Dynamics in Rydberg Atom Quantum Simulators

Gauge theories are the cornerstone of our understanding of fundamental interactions among particles. Their properties are often probed in dynamical experiments, such as those performed at ion colliders and high-intensity laser facilities. Describing the evolution of these strongly coupled systems is a formidable challenge for classical computers, and represents one of the key open quests for quantum simulation approaches to particle physics phenomena. Here, we show how recent experiments done on Rydberg atom chains naturally realize the real-time dynamics of a lattice gauge theory at system sizes at the boundary of classical computational methods. We prove that the constrained Hamiltonian dynamics induced by strong Rydberg interactions maps exactly onto the one of a $U(1)$ lattice gauge theory. Building on this correspondence, we show that the recently observed anomalously slow dynamics corresponds to a string-inversion mechanism, reminiscent of the string-breaking typically observed in gauge theories. This underlies the generality of this slow dynamics, which we illustrate in the context of one-dimensional quantum electrodynamics on the lattice. Within the same platform, we propose a set of experiments that generically show long-lived oscillations, including the evolution of particle-antiparticle pairs. Our work shows that the state of the art for quantum simulation of lattice gauge theories is at 51 qubits, and connects the recently observed slow dynamics in atomic systems to archetypal phenomena in particle physics