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Simulating 2D lattice gauge theories on a qudit quantum computer

Michael Meth, Jan F. Haase, Jinglei Zhang, Claire Edmunds, Lukas Postler, Alex Steiner, Andrew J. Jena, Luca Dellantonio, Rainer Blatt, Peter Zoller, Thomas Monz, Philipp Schindler, Christine Muschik, Martin Ringbauer

arXiv:2310.12110v3quant-ph

TL;DR

Classical lattice-gauge calculations are impeded by sign problems in settings such as real-time dynamics and finite matter density. This work uses a qudit quantum processor to simulate 2D-QED, measuring a single plaquette and examining gauge-field truncations. It reports ground-state effects of dynamical matter and qudit resource advantages, and points toward simulations of higher-dimensional lattice gauge theories.

  • Problem

    Classical Monte Carlo lattice-gauge calculations break down for problems including finite matter density and real-time dynamics, motivating alternative computational methods.

    Source: Appendix F · p. 17

  • Method

    The work simulates a single 2D-QED plaquette on a qudit quantum computer and demonstrates changing the gauge-field discretization from qutrits to ququints.

    Source: §I · p. 2, §IV · p. 4

  • Results

    The plaquette study observes ground-state effects of dynamical matter, while qudit encodings use fewer entangling gates, smaller registers, and improve simulation accuracy relative to conventional qubit devices.

    Source: §I · p. 2

  • Takeaways & Limitations

    The protocol for eliminating redundant gauge degrees of freedom can be extended to 3D QED, where gauge fields must be simulated explicitly.

    Source: §VI · p. 7

Abstract

from arXiv · show

Particle physics underpins our understanding of the world at a fundamental level by describing the interplay of matter and forces through gauge theories. Yet, despite their unmatched success, the intrinsic quantum mechanical nature of gauge theories makes important problem classes notoriously difficult to address with classical computational techniques. A promising way to overcome these roadblocks is offered by quantum computers, which are based on the same laws that make the classical computations so difficult. Here, we present a quantum computation of the properties of the basic building block of two-dimensional lattice quantum electrodynamics, involving both gauge fields and matter. This computation is made possible by the use of a trapped-ion qudit quantum processor, where quantum information is encoded in $d$ different states per ion, rather than in two states as in qubits. Qudits are ideally suited for describing gauge fields, which are naturally high-dimensional, leading to a dramatic reduction in the quantum register size and circuit complexity. Using a variational quantum eigensolver, we find the ground state of the model and observe the interplay between virtual pair creation and quantized magnetic field effects. The qudit approach further allows us to seamlessly observe the effect of different gauge field truncations by controlling the qudit dimension. Our results open the door for hardware-efficient quantum simulations with qudits in near-term quantum devices.

I. INTRODUCTION

Classical lattice-gauge simulations face sign problems in important regimes, while earlier quantum demonstrations were limited to one dimension or theories with trivial gauge fields or matter. This work addresses simulation beyond one dimension with both fields and matter, and control over gauge-field dimension.

I. INTRODUCTION

  • The work realizes programmable, high-fidelity entangling gates for mixed-dimensional qudit systems on an all-to-all connected linear ion-trap processor.The paper describes native d-level simulation with smaller registers and reduced gate depth compared with a qubit approach.Source: §I · p. 1
  • Classical lattice-gauge simulations encounter sign problems in important regimes, and prior quantum demonstrations were limited to 1D or theories with trivial gauge fields or matter.The stated problem areas include real-time evolution and high matter densities.Source: §I · p. 1
  • Gauge-theory calculations beyond 1D require finite truncations of formally infinite-dimensional gauge fields that remain large enough to capture relevant physics.The paper identifies both simulation beyond 1D with gauge fields and matter and control of gauge-field dimension as challenges.Source: §I · pp. 1–2
  • The experiment simulates a single 2D-QED plaquette and measures its ground-state plaquette expectation value, relevant to magnetic fields and the running coupling.The paper reports an order-of-magnitude improvement in circuit complexity in the simplest instance.Source: §I · p. 2

II. LGT SIMULATIONS WITH QUDITS

The section describes how the model’s matter and gauge fields are represented, how physical states are constrained, and why direct qudit encoding suits truncated gauge fields. It also outlines the trapped-ion implementation and variational ground-state search.

II. LGT SIMULATIONS WITH QUDITS

  • The model’s Hamiltonian includes electric, magnetic, and matter-field energies, plus a kinetic term responsible for pair creation.The bare coupling and mass set the energy cost of creating a particle pair and its associated electromagnetic field; Ω characterizes their creation rate.Source: §II · p. 2
  • Physical states must satisfy Gauss’ law at each lattice site, expressed as a constraint on the gauge-field and charge configurations.The Gauss operator annihilates physical states.Source: §II · p. 2
  • The plaquette operator measures a closed-loop gauge-field quantity related to the curl of the vector potential and to the running coupling.Its ground-state expectation value varies with g−2 and has no analogue in 1D-QED.Source: §II · p. 2
  • Gauge-field spectra require truncation to at least three states, while direct d-level qudit encoding keeps gauge-field operations local instead of decomposing them into complex qubit interactions.The text states that qubit encodings require at least ⌈log2(d)⌉ qubits and that local gauge-field operators involve O(d2) two-qubit gates.Source: §II · p. 2
  • Trapped 40Ca+ ions encode gauge fields in qudit levels and matter in qubits, with qubit-qudit interactions realized through entanglement with a shared motional mode.The ion’s internal state conditionally injects a phonon, which controls a local operation on the second ion before the phonon is removed.Source: §II · p. 3, Figure 1 · p. 3
  • A classical optimizer varies parameterized circuit gates to minimize the measured energy, yielding an ansatz state that approximates the ground state.The figure describes a mixed-dimensional circuit whose qudit is coupled to selected subspaces.Source: §II · pp. 3–4, Figure 1 · p. 3

III. SIMULATING GAUGE FIELDS AND MATTER

The single-plaquette experiment measures the ground-state plaquette expectation value and agrees with theoretical predictions. In the intermediate coupling regime, dynamical matter and pair creation increase that expectation value.

III. SIMULATING GAUGE FIELDS AND MATTER

  • Gauss’ law eliminates redundant gauge degrees of freedom, reducing the single open-boundary plaquette Hamiltonian to one gauge field while guaranteeing physical states.The minimal d = 3 truncation uses one qutrit for that field and four qubits for the matter sites.Source: §III · p. 4
  • The variational circuit initializes the matter and gauge fields in the bare vacuum, then uses matter gates and controlled qubit-qutrit rotations to represent pair creation with corresponding flux changes.The final matter gates adjust gauge-field-independent free dynamics.Source: §III · p. 4
  • The measured plaquette expectation values approximate the ideal result and agree with theoretical predictions; dynamical matter increases the value in the intermediate coupling regime.The plotted experimental points are compared with exact-diagonalization results and an experimentally motivated noise model.Source: §III · p. 4
  • At large coupling the electric term favors a zero plaquette expectation value, while at weak coupling the magnetic term favors approximately 0.707 for the chosen truncation.Near g−2 ≈ 1, competition between field energies and the kinetic term makes pair creation energetically favorable; the resulting matter-induced flux increases the plaquette expectation value.Source: §III · p. 4

IV. TOWARDS REFINING THE GAUGE FIELD DISCRETIZATION

The section tests how gauge-field truncation and basis choice affect a pure-gauge 2D-QED plaquette, using a torus represented by three rotators. Its results show that representation performance depends on coupling and that increasing the qudit dimension reduces truncation effects.

IV. TOWARDS REFINING THE GAUGE FIELD DISCRETIZATION

  • The minimal periodic system has four vertices and eight gauge links, and Gauss’ law leaves three rotators sufficient to describe ground-state properties.The simulation uses those rotators to represent circulation paths around plaquettes.Source: §IV · p. 5, Figure 2 · p. 5
  • The truncation scheme chooses an electric-field basis when the electric Hamiltonian dominates and a magnetic-field basis when the magnetic term dominates.The electric basis is suited to g^-2 ≪ 1, while the magnetic basis is suited to g^-2 ≫ 1.Source: §IV · p. 5
  • As the truncation increases, electric- and magnetic-representation plaquette curves approach each other and agree near an intermediate coupling, indicating where to switch representations.The d = 21 numerical result supports the approach to agreement.Source: §IV · p. 6
  • In the weak-coupling regime, a broad electric-field distribution makes electric-basis truncation increasingly costly, while the magnetic representation behaves complementarily.The closing gap between representation curves at intermediate coupling indicates how well finite-d computations approximate the untruncated result.Source: §IV · p. 6
  • Premature flattening of the plaquette expectation value at weak coupling signals that the gauge-field dimension is too small to represent the changing ground state accurately.The cited analysis reports fast convergence of the plaquette expectation value for d ≤ 10.Source: §IV · p. 6
  • Increasing the discretization from qutrits to ququints requires only minor experimental changes, with the same ions and entangling gates.The circuit comparison is shown for qutrit and ququint truncations.Source: Figure 2 · p. 5, §IV · p. 6

V. TOWARDS REAL-TIME DYNAMICS

The section takes a first step toward real-time simulation by applying a Trotter protocol to a plaquette with gauge fields and dynamical matter. Following a quench, particle density rises and then oscillates as pairs are created and annihilated, with corresponding plaquette dynamics.

V. TOWARDS REAL-TIME DYNAMICS

  • The digital simulation uses mixed-dimensional entangling gates in a Trotter protocol for a plaquette with open boundaries, gauge fields, and dynamical matter.The study starts from the bare vacuum and varies the length of a single Trotter step.Source: §V · p. 6
  • The kinetic term drives particle-antiparticle creation and subsequent annihilation after the quench, producing an increase and then oscillations in mean particle density.Gauss’ law links the charged-particle dynamics to corresponding changes in the plaquette expectation value.Source: §V · p. 7
  • The evolution is a quench from the strong-coupling regime, where the bare vacuum is the ground state, to an intermediate coupling.The kinetic term then drives the pair dynamics.Source: §V · pp. 6–7

VI. OUTLOOK

The outlook presents qudits as a hardware-efficient route to simulating lattice gauge theories beyond one dimension, including explicit gauge fields and dynamical matter. It identifies scaling system sizes as the remaining major task on the path toward three-dimensional simulations.

VI. OUTLOOK

  • A universal trapped-ion qudit processor with all-to-all connectivity supports arbitrary geometries and a 2D-QED building block with dynamical matter and gauge fields.The work also studies different gauge-field discretizations.Source: §VI · p. 7
  • The VQE circuits used for equilibrium properties are much shallower than comparable qubit-based gauge-theory implementations, and the techniques are adaptable to digital real-time simulations.These are presented as features of the demonstrated qudit approach.Source: §VI · p. 7
  • Scaling up system sizes remains the major task for extending these simulations toward three spatial dimensions.The protocol for eliminating redundant gauge degrees of freedom is described as directly extendable to 3D QED.Source: §VI · p. 7

Appendix A: Realizing Controlled Rotations in Qudits

This appendix outlines trapped-ion qudit implementation of conditional rotations and the simulation methods, including model specifications, encodings, circuits, optimization, and measurements. It also describes Hamiltonian operator decomposition and generalized Clifford gates for simultaneous measurement of a subset.

Appendix A: Realizing Controlled Rotations in Qudits

  • Conditional rotations use control-state-dependent phonon creation, three blue-sideband pulses on the target, and reversed control operations to disentangle the qudits from motion.The axial center-of-mass mode provides homogeneous coupling; AC Stark shifts are compensated with an off-resonant beam and software frame updates.Source: Appendix A · p. 11, Figure 5 · p. 12, Figure 6 · p. 12

Appendix B: 2D-QED Hamiltonian

  • Physical states obey Gauss’ law with zero eigenvalues, and the study restricts the conserved total charge to zero.The plaquette operator represents magnetic-field energy, which has no analogue in one-dimensional QED.Source: Appendix B · p. 13

Appendix C: Encoded Hamiltonian

  • Appendix C introduces the Hamiltonians studied with the VQE circuits shown in Figs. 1d and 2c.Source: Appendix C · p. 13

1. 2D-QED with Matter

  • For the open-boundary matter model, Gauss’ law eliminates three gauge fields, leaving four fermionic fields and one gauge field; the gauge degree of freedom is encoded in a qutrit.The experiment uses m = 0.1 and Ω = 5 to enter a nonperturbative regime where pair creation significantly affects the model.Source: §1 · pp. 13–14

2. Pure Gauge 2D-QED

  • The periodic pure-gauge model has four vertices and reduces by Gauss’ law to three independent plaquette degrees of freedom represented by integer-spectrum rotators.Truncation uses 2l + 1 states per rotator; the calculations use (L, l) = (2, 1) for qutrits and (3, 2) for ququints.Source: §2 · p. 14

1. Variational Circuit

  • The variational circuit uses real-coefficient gates, including magnetization-conserving qubit gates and controlled rotations between addressed qudit levels.The controlled rotations are active for a specified control state, while qudit-internal rotations act between selected levels.Source: §1 · p. 15

2. Optimization Algorithm

  • Both VQE optimization strategies use Bayesian optimization with Gaussian-process surrogates, avoiding gradient evaluations and tolerating noisy cost measurements.For matter and gauge fields, stored outcomes are reused across coupling values; the noisy trust-region strategy performs two sweeps from g−2 = 0.01 to g−2 = 100.Source: §2 · p. 15

3. VQE Measurements for Qudits

  • For qudit VQE measurements, the Hamiltonian is represented using clock and shift operators normalized by the generalized Clifford group.This addresses the lack of an evident optimal decomposition basis for qudit Hamiltonians, unlike the Pauli-product approach for qubits.Source: §3 · pp. 15–16
  • The Hamiltonian is expressed as a sum of operators whose indices are determined by its decomposition.Its contributions are written in terms of clock and shift operators.Source: §3 · p. 16
  • After normalization with generalized Clifford gates, a subset of the operators maps to clock operators Z_γi that can be measured simultaneously.The mapped operators need not share the same γi.Source: §3 · p. 16

Appendix E: Real-time Evolution

The appendix demonstrates Trotter-simulated real-time evolution of a single open plaquette with dynamical matter, then places quantum simulations in the context of problems inaccessible to MCMC. It identifies higher-dimensional and non-Abelian gauge theories as further challenges.

Appendix E: Real-time Evolution

  • A one-step Trotter protocol simulates a quench from the bare vacuum at strong coupling to weaker coupling on an open plaquette with dynamical matter.The initial bare vacuum contains no particles or gauge-field excitations.Source: Appendix E · p. 16, Figure 8 · p. 17
  • After the quench, pair creation raises particle density, while subsequent pair annihilation lowers it; Fig. 3 compares Trotter approximations of particle density and plaquette expectation value.The comparison includes N_T = 1, N_T = 10, and the exact result as N_T → ∞.Source: Appendix E · p. 17

Appendix F: Quantum Computing for Particle Physics

  • MCMC-based lattice gauge simulations are limited for finite fermionic chemical potentials and real-time dynamics, motivating quantum-computing approaches.The appendix connects these challenges to the QCD phase diagram at finite densities and real-time collider evolution.Source: Appendix F · p. 17
  • The appendix identifies 3D simulations requiring at least three gauge levels and non-Abelian simulations including color degrees of freedom as key challenges.It frames both as steps for applying quantum computers to the problem areas discussed.Source: Appendix F · p. 17

Appendix G: Qudit Advantage

The appendix compares native qudit and qubit implementations for full and pure-gauge plaquettes, using CNOT gate counts and device-specific runtime estimates. The qudit approach has more favorable gate-count scaling, while its longer readout still leaves the runtime tradeoff slightly in its favor for the studied problems.

Appendix G: Qudit Advantage

  • The comparison assumes embedded qudit CNOT gates have roughly the same fidelity as the corresponding gates in a pure qubit system.The authors state this assumption applies to their trapped-ion platform.Source: Appendix G · p. 17
  • 18 + 4(d −1) versus 18 + 36(d −1) CNOT gates: the qudit circuit scales about an order of magnitude better for a full 2D plaquette.The comparison uses CNOT-based counts for cross-platform comparability; Table I describes the full gauge-theory plaquette.Source: Appendix G · p. 18, Table I · p. 18
  • For a pure-gauge plaquette, qudit gate count does not scale with truncation, whereas the qubit count scales as 48 + 12d.A register-size-optimized qubit encoding reduces register growth to 3⌈log2(d)⌉, but its gate count generally scales quadratically with dimension.Source: Appendix G · p. 18, Table II · p. 18
  • The qudit readout versus qubit-gate runtime tradeoff slightly favors qudits for the problems studied, despite generally longer qudit circuits from readout.The comparison uses qudit readout of (d −2)4 ms and qubit gates at 0.2 ms per gate; Table III's runtime estimates are for the device used in the main text.Source: Appendix G · p. 18

Appendix H: VQE Circuit for Pure-Gauge QED with Periodic Boundary Conditions

The appendix describes pure-gauge VQE circuits designed around the Hamiltonian and the coupling structure between rotators. It uses symmetry to reduce independent parameters and gives a separate magnetic-representation circuit.

Appendix H: VQE Circuit for Pure-Gauge QED with Periodic Boundary Conditions

  • The pure-gauge VQE circuits are inspired by the Hamiltonian and provide a strategy intended to scale to higher gauge-field truncations.Separate variational circuits are used for the electric and magnetic representations.Source: Appendix H · p. 18
  • In the electric representation, superpositions for rotator R2 are followed by controlled rotations reflecting the coupling between rotators.Symmetry between R1 and R3 reduces the number of independent variational parameters.Source: Appendix H · p. 19
  • For larger g^-2, the circuit spreads rotator populations 1 and 3 by controlling on the corresponding state of rotator 2.This design addresses the tendency for all three rotators to occupy the same electric-field eigenstate.Source: Appendix H · p. 19
  • The magnetic-representation circuit is designed using arguments analogous to those for the electric Hamiltonian.Figure 10 specifies the magnetic-representation circuit for qutrit and ququint truncations.Source: Appendix H · p. 19, Figure 10 · p. 19

Appendix I: Experimental Details

The experimental details describe encoding choices for reducing cross-talk and optimizing trapped-ion operations, gate-count shortcuts, and sequential qudit measurement and readout.

1. Optimal Encoding and Cross-Talk Suppression

  • The ququint encoding selects states with strong blue-sideband coupling and restricts certain transitions to ±π rotations to make Stark shifts constant and easier to compensate.For qutrits, neighboring ions can use different Zeeman-state sets to suppress next-neighbor cross-talk; that spectroscopic scheme is unavailable for ququints in the stated ion manifold.Source: §1 · pp. 19–20, Figure 11 · p. 19, Figure 12 · p. 20

2. Gate Count Optimization

  • Known input states reduce a controlled rotation from five blue-sideband pulses to two, while successive rotations on the same control state can cancel and selected SWAPs use three MS gates.These shortcuts are applied to specified matter-qubit operations and SWAPs in the circuit.Source: §2 · p. 20

3. Measurement and Readout of Qudits

  • Qudit measurement reorders states into a W pattern for QR-decomposition-based compilation, followed by d −1 detection events and local π rotations.Readout uses d −2 local π flips and d −1 detection events, with duration scaling approximately as (d −2) · 3.5 ms + 1 ms.Source: §3 · pp. 20–21, Figure 14 · p. 20

Appendix J: Noise Model

The noise model reflects gate-angle fluctuations and phase errors in trapped-ion qudit operations, with simplifying assumptions used for spectator phases. Simulated noisy VQE data agrees qualitatively with experiment.

Amplitude Fluctuations

  • The model includes stochastic gate-angle fluctuations, using tenfold smaller variance for single-qudit gates than for parametrized entangling gates.The angle fluctuation is Gaussian with zero mean.Source: Appendix J · p. 21

Phase Errors

  • Phase errors model AC Stark shifts: the driven-state phase is neglected, while passively compensated spectator levels share a common phase Φ.The common-phase assumption simplifies the phase mappings used in the noise model.Source: Appendix J · p. 21
  • Consecutive C-ROT gates with the same control qudit and state can be contracted, reducing the number of blue-sideband pulses.In this case, the control qudit acquires phase on its spectator levels.Source: Appendix J · p. 21

Results

  • Experimental results show good qualitative agreement with simulated noisy VQE data using noise parameters estimated at σ²p = 0.073.The simulation uses variational parameters obtained without noise before evaluating noisy circuits.Source: Appendix J · p. 21

Appendix K: Results for Alternative Lattice QED Model

This appendix studies an alternative compact lattice QED model and reports VQE plaquette measurements that agree with theory. Its intermediate-coupling behavior reflects competition between field-energy and kinetic terms, including a pair-creation-associated dip.

Appendix K: Results for Alternative Lattice QED Model

  • The two compact-QED models differ in the kinetic-term phase and continuum limit, but coincide in the pure-gauge case.For the pure-gauge model, the appendix describes truncations using qutrit and ququint representations.Source: Appendix K · p. 22, Figure 15 · p. 22
  • Measured ground-state plaquette expectation values for the alternative model agree with theoretical predictions at Ω = 5 and m = 0.1.The experiment uses the same VQE circuit structure as the earlier model.Source: Appendix K · p. 22, Figure 16 · p. 23
  • At weak coupling the magnetic term favors a positive vacuum plaquette value, whereas at large coupling the electric term favors a zero value.Near g^-2 ≈ 1, the electric and magnetic terms compete with the kinetic term.Source: Appendix K · p. 22
  • The plaquette expectation value shows a dip attributed to pair creation in the ground state, even for the small system studied.The kinetic term makes pair creation energetically favorable and produces background magnetic flux opposing the flux favored by the magnetic-field term.Source: Appendix K · pp. 22–23
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