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Quantum-Classical Computation of Schwinger Model Dynamics using Quantum Computers

N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris, R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, M. J. Savage

arXiv:1803.03326v3quant-phhep-lathep-phhep-thnucl-th

TL;DR

The paper addresses how to study lattice gauge-theory dynamics on quantum hardware limited by qubit counts, noise, and coherence. It combines classical symmetry-sector identification with quantum evaluation of Schwinger-model dynamics, reducing the relevant Hilbert space and enabling IBM-hardware calculations. The approach is presented as a step toward applying NISQ quantum computers to broader theories such as QCD.

  • Problem

    Quantum field-theory calculations can be inaccessible with conventional computing, while near-term quantum hardware has limited qubits, noisy gates, and coherence times.

  • Method

    The paper uses classical computation to identify Gauss-law and momentum/parity symmetry sectors, then evaluates dynamics in the reduced sectors on a digital quantum computer.

  • Results

    The method softens exponential Hilbert-space growth and enables static and dynamic Schwinger-model observables on IBM superconducting quantum hardware.

  • Takeaways & Limitations

    The work is one step toward solving QCD and other lattice quantum field theories with NISQ-era quantum computers.

Abstract

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We present a quantum-classical algorithm to study the dynamics of the two-spatial-site Schwinger model on IBM's quantum computers. Using rotational symmetries, total charge, and parity, the number of qubits needed to perform computation is reduced by a factor of $\sim 5$, removing exponentially-large unphysical sectors from the Hilbert space. Our work opens an avenue for exploration of other lattice quantum field theories, such as quantum chromodynamics, where classical computation is used to find symmetry sectors in which the quantum computer evaluates the dynamics of quantum fluctuations.

I. INTRODUCTION

The paper develops a hybrid quantum-classical strategy for studying Schwinger-model ground states and real-time dynamics on IBM quantum hardware. Symmetry-sector projections reduce the calculation to physically allowed states despite NISQ hardware constraints.

  • The work targets low-energy quantum field-theory problems that are difficult for conventional computation and constrained on NISQ hardware by limited qubits, connectivity, and noisy gates.
  • The authors develop a hybrid strategy for the lattice Schwinger model and study particle and electric-field energy-density dynamics alongside ground-state properties.
  • Periodic boundary conditions provide discrete rotational and reflection symmetries, enabling classification by momentum, charge, and parity.
  • Classical computation identifies physical symmetry sectors, while a digital quantum computer evaluates dynamics within each sector using one- and two-qubit gates.
  • The study demonstrates calculations for one- and two-spatial-site Schwinger models on IBM quantum computers.

II. THE SCHWINGER MODEL

The Schwinger model is a one-space, one-time-dimensional gauge theory discretized into fermion sites and quantized electric-flux links. For two spatial sites, Gauss’s law and symmetry projections remove most unphysical states and reduce the computational representation.

  • The Schwinger model serves as a prototype for strong interactions, sharing confinement, chiral-symmetry breaking, charge screening, and a nonzero vacuum condensate with QCD.
  • Kogut-Susskind discretization places electron and positron components on neighboring even-odd sites, producing two fermion sites per spatial site.
  • Each spatial site uses two fermion-occupation qubits, while electric-flux links have integer-valued flux and raising and lowering operators.
  • For two spatial sites, 1296 states fit in 12 qubits, but Gauss’s law isolates 13 physical Q = 0 states before momentum and parity projection.
  • The naïve 12-qubit description is reduced to 3 qubits through physical-state and symmetry-sector reduction, with a further electric-field-energy cutoff available.
  • On current hardware, state reduction introduces approximately 1% systematic error in low-lying energies and requires classical computation that becomes exponentially costly with system size.

III. GROUND STATE CALCULATIONS

Ground-state calculations use VQE with classical Bayesian optimization and error mitigation on IBM hardware. The reported energies stabilize across the tested electric-field cutoffs, while the computation requires substantial wall-clock communication time.

  • VQE with classical Bayesian global optimization extracts the ground-state energy in the P = +1 sector using few quantum-computer function calls.
  • The P = +1 Hamiltonian has a one-dimensional chain structure with N = ˜Λ + 1 sites and nearest-neighbor hopping.
  • ⟨H⟩ = −0.91(1) MeV, −1.01(4) MeV, and −1.01(2) MeV for ˜Λ = 1, 2, 3 respectively.
  • The calculations use 8192 measurement shots, readout-error mitigation, and zero-noise extrapolation based on inserting additional CNOT gates.
  • The VQE calculation used 6.4 QPU-seconds and 2.4 CPU-seconds but had a total runtime of 4 hours, mostly spent in communication.

IV. DYNAMICAL PROPERTIES

The study evolves the k = 0, P = +1 sector from the unoccupied state using SU(4) parameterization and first-order Trotterization, comparing quantum-hardware results with exact dynamics. Trotterization preserves coherence for only about 10 time steps on ibmqx2, creating an accuracy–coherence trade-off.

  • Time-evolution methods: The unoccupied state |χ1⟩k=0,+ is evolved with SU(4) parameterization and Trotter discretization on IBM hardware.The SU(4) approach uses classical preprocessing to determine nine time-evolution angles.
  • SU(4) evolution: The SU(4) implementation produces zero-noise-extrapolated pair probability and electric-field energy as functions of time using a three-CNOT circuit.The data points are quadratic extrapolations, while solid curves provide exact results.
  • Trotter evolution: The Trotterized operator UT(t, δt) provides an alternative time-evolution method that avoids solving classically for nine SU(4) angles.Its advantage in classical resource requirements is offset by greater demands on hardware coherence.
  • Hardware limitation: Approximately 10 coherent Trotter time steps are available before ibmqx2 reaches its T2 coherence time.For δt = 0.1, the pair probability saturates to the classical value 0.5 after coherence is lost.
  • Hardware limitation: Larger δt values extend maintained coherence but trade away Trotterization accuracy and still do not reach the low-energy structure of dynamic fluctuations.The trade-off is illustrated by the shaded-region data in Fig. 4.

V. DISCUSSION AND OUTLOOK

The discussion identifies classical symmetry-sector reduction as essential for near-term quantum simulations of the Schwinger model and as a step toward more complex lattice gauge theories. It also outlines the symmetry structure used for the two-site system and the need to balance circuit depth against classical preprocessing in future work.

  • Future development: Short-depth exact propagators and manageable-resource Trotterization must be balanced to study scattering and charge-screening dynamics on imperfect quantum computers.The paper identifies this balance as a key area for future development.
  • Hybrid framework: Classical preprocessing enforces Gauss’s law, projects momentum, and imposes parity, reducing the Hilbert-space growth enough for calculations on IBM hardware.The framework combines classical Hilbert-space reduction with quantum evolution of the projected dynamics.
  • Hybrid framework: The framework does not provide quantum advantage in accessible Hilbert-space dimensionality because the physical projected basis is constructed classically.Its proposed advantage is instead framed as multidimensional and based on combining classical and quantum strengths.
  • Symmetry sectors: The two-site Q = 0 system contains 13 Gauss-law-compatible states that can be organized into momentum and parity sectors.Momentum projection uses translation by one spatial site, after which parity combinations form distinct sectors.
  • Discrete symmetries: For the two-site lattice, reflections preserve the discretization about electron-defined and positron-defined axes, while charge conjugation exchanges particles and antiparticles and reverses electric-field direction.Charge conjugation also requires a half-site shift to preserve the qubit structure.
  • Outlook: The parity construction extends naturally to lattices with an even number of spatial sites, including Nsites = 4, 6, 8, ....This provides the stated route for extending the symmetry analysis beyond two spatial sites.

Appendix B: Exact two-site Schwinger Model Spectra

The two-site Schwinger model spectrum contains parity-resolved low-lying states, while finite-volume effects shape excited-state splittings and vacuum-property convergence.

  • Spectrum: The two-site spectrum uses μ = 0.1, x = 0.6, electric-field cutoff Λ̃ = 10, and zero-momentum projection.The ground-state energy is shifted to zero in the displayed spectrum.
  • Spectrum: 0, 2.089, and 3.108 are the shown shifted P = +1 energy eigenvalues, while P = −1 eigenvalues are 1.497 and 2.927.
  • Spectrum: The first excited state is odd parity and identified as the lightest vector meson, while the second is even parity and identified as the scalar meson.
  • Finite-volume effects: The next even-parity state lies just above the V−V− threshold, with its splitting attributed to a finite-volume effect that vanishes at infinite volume.
  • Finite-volume effects: Vacuum energy density rapidly converges and is within ∼1% of its infinite-volume value with two spatial sites for the chosen parameters.The vacuum energy itself is extensive, whereas the energy density approaches a constant.

Appendix C: SU(4) Transformations for 2-qubits

The appendix parameterizes two-qubit SU(4) transformations through local SU(2) rotations and Cartan-subalgebra operations, exploiting symmetry to reduce parameters and comparing circuit realizations.

  • SU(4) parameterization: Two-qubit unitary rotations belong to SU(4), whose general Pauli-basis parameterization requires 15 real angles.
  • SU(4) parameterization: Symmetry reduces the SU(4) parameter count from 15 to 9, and removing relative eigenbasis phases reduces it further to 6.The six-angle reduction applies to the symmetric transformations used for time evolution and variational state preparation.
  • Decomposition: The decomposition uses K1 and K2 from SU(2)⊗SU(2), acting on individual qubits, together with Cartan-subalgebra transformations C.
  • Decomposition: The local SU(2) factors are parameterized with standard ZYZ Euler angles, with explicit circuit forms given for the symmetric two-qubit transformation.
  • Hardware implementation: 3-CNOT and 6-CNOT implementations are technically equivalent in simulation, but hardware systematic errors are exacerbated at high probabilities for the 6-CNOT circuit.
  • Hardware implementation: Figure 7 compares one-e+e−-pair probabilities after one exact-propagator application on IBM simulators and ibmqx2 hardware using both circuit constructions.

Appendix D: Trotterization

The appendix develops Trotterized time evolution for near-term hardware and variational circuits for the reduced Schwinger-model Hamiltonian. It emphasizes balancing Trotter errors with hardware constraints and exploits the Hamiltonian’s structure to reduce variational costs.

  • Trotterization: Exact propagators become impractical as qubit count grows because their angle parameter space scales exponentially with system size.For n qubits, the parameter-space dimensionality is 2^(n−1)(2^n+1)−1.
  • Trotterization: First-order Trotterization approximates evolution by splitting the Hamiltonian exponential into NTrot. time steps and suppressing commutator terms in powers of Hδt.The total propagation time is divided into NTrot. steps.
  • Trotterization: Near-term IBM hardware requires balancing Trotterization errors against gate fidelities and coherence times.The authors report that sub-leading terms are not yet negligible in IBM simulations of the Schwinger model.
  • Trotterization: Permutation-order sampling provides a rudimentary optimization of the Trotterized propagator for the 4×4 k = 0, P = +1 Hamiltonian sector.The sampled orderings are evaluated through propagator differences and e+e− pair probabilities.
  • Variational calculations: The reduced Hamiltonian’s nearest-neighbour structure permits variational ground- and excited-state calculations using few rotation angles and comparatively low cost.The ground-state ansatz uses three angles, while a similar two-angle form can construct the first excited state.
  • Variational calculations: Bayesian optimization refines angle estimates after an initial grid sampling, while operator expectation values provide the energy for minimization.The procedure uses approximately 10 initial angle sets and then a second iteration to determine the ground-state energy precisely.

Appendix F: Convergence with the cut-off in the gauge-field energy

The appendix examines how truncating electric-field energy affects spectra and dynamics. Ground-state energies converge at relatively low cutoffs, but dynamical pair fluctuations require stricter convergence checks.

  • Energy-cutoff convergence: The reduced Hilbert space is obtained by sequentially removing the highest total-electric-field-energy states from the basis.The cutoff ˜Λ controls the total retained electric-field energy, while Λ bounds the energy on each flux link.
  • Energy-cutoff convergence: Removing the highest-energy state at ˜Λ = 3 introduces less than 1% systematic error in the P = +1 ground-state energy.This cutoff corresponds to the reduced two-qubit form of the sector.
  • Energy-cutoff convergence: Dynamical pair fluctuations change significantly with increasing ˜Λ when few states are retained, whereas the dynamics become stable beyond Λ = 2.Figure 9 compares truncated dynamics with the untruncated ˜Λ = 4 result and shows early convergence in Λ.
  • Energy-cutoff convergence: A cutoff that accurately reproduces the ground-state energy can still fail to converge the dynamics, even when Λ = 1 and ˜Λ is unrestricted.The ground-state energy is a relatively weak constraint on the full wavefunction.
  • Variational implications: Lower energy cutoffs reduce the number of variational angles and provide perturbatively close Bayesian priors for subsets of angles at larger cutoffs.The authors state that this hierarchy was explicitly verified.

Appendix G: Scaling to Larger Lattices

The paper projects the lattice Hilbert space onto physical, zero-momentum, and definite-parity sectors to obtain exponential reductions in the relevant state-space dimension. These reductions remove unphysical or symmetry-disconnected sectors while qubit requirements still grow linearly with lattice size.

  • Hilbert-space scaling: Gauss’s law reduces the Hilbert-space scaling from e^(log(64)Ns) to 1.02(1)e^(1.1772(2)Ns).The unreduced scaling corresponds to one qubit per site and two qubits per link at Λ = 1.
  • Hilbert-space scaling: Projecting further to k = 0 and even parity reduces the scaling to 0.29(5)e^(1.006(23)Ns).Each reduction removes an exponentially large unphysical or symmetry-disconnected contribution.
  • Reduction procedure: The number of required qubits grows linearly with system size both before and after the reductions.Combinatoric calculations reach the physical-space scaling at and beyond 80 spatial sites, while additional global-symmetry counting remains more complex.
  • Reduction procedure: The lattice-to-reduced mapping proceeds through Gauss-law constraints, zero-momentum projection, and definite-parity projection.Table III organizes these reductions from the naturally defined lattice representation to the final symmetry sector.

Appendix H: Quantifying the CNOT systematic errors

The appendix characterizes CNOT-gate systematics through noise-parameter sweeps and extrapolation, while identifying limits of applying the procedure to variational ground-state searches and large gate depths.

  • CNOT error extrapolation: Applying a CNOT gate r times yields a linear relation between noisy and noiseless Pauli-operator expectation values for small gate error.The analysis uses CNOT^2 = I and a white-noise channel before fitting temporal data linearly or quadratically.
  • Dynamic evolution: Quadratic extrapolation was applied to reduce CNOT-gate systematic errors in the time-dependent pair-density calculation.Figure 10 compares different CNOT counts, extrapolated red points, and the exact gray curve using 8K measurements per point.
  • Variational ground states: Ground-state extrapolation is biased because optimizing angles at r = 1 and removing noise bias do not commute.The extrapolated r = 0 hypersurface does not directly produce the r = 0 ground state when angles remain optimized at r = 1.
  • Variational ground states: Interchanging minimization and extrapolation would increase the variational cost by roughly a factor of 4 while removing the CNOT bias from the ground-state wavefunction.This requires Bayesian optimization on the r = 0 hypersurface rather than optimization on the original r = 1 circuit.
  • Large-depth behavior: Linear and quadratic extrapolations are reliable only while CNOT systematic errors remain small; larger depths produce nonlinear or oscillatory behavior.At large gate counts, observables approach an r-independent classical regime as the density matrix tends toward the identity.
  • Chiral condensate: The variational calculation obtained a chiral condensate of −0.296(13), consistent with the exact known result.The condensate varies more strongly near the energy minimum than the energy because it probes different ground-state attributes.

Appendix K: Data Tables

The appendix collects numerical data underlying the paper’s energy, condensate, pair-density, electric-field, and CNOT-noise figures.

  • Data tables: Table IV contains the energy and chiral-condensate data shown in Fig. 2.
  • Data tables: The listed pair-density datasets include values across scaled time and CNOT noise parameters r = 1, 3, 5, 7, with extrapolated estimates.
  • Data tables: The appendix lists pair-density data for the main-text figures and CNOT-noise analyses.These include Figs. 3, 4, 7, and 10, plus the noise-parameter data associated with Fig. 11.
  • Data tables: The electric-field energy data shown in Fig. 3 are provided separately in Table VI.
  • Data tables: The appendix tabulates zero-pair probabilities at two times as a function of the noise parameter for Fig. 12.
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