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General Methods for Digital Quantum Simulation of Gauge Theories

Henry Lamm, Scott Lawrence, Yukari Yamauchi

arXiv:1903.08807v3hep-latcond-mat.str-elnucl-thquant-ph

TL;DR

Classical nonperturbative simulation and gauge-theory implementation pose challenges for quantum-field-theory simulation. The paper constructs gauge-preserving Trotter evolution and gauge-invariant time-separated correlators, then demonstrates the procedure in two simulated gauge theories.

  • Problem

    Classical methods face exponential state spaces and sign problems, while gauge-field representation and time-separated composite observables present additional simulation challenges.

    Source: §I · p. 1

  • Method

    The paper maps gauge-theory evolution to a quantum computer with a gauge-invariant Trotterized operator and constructs gauge-invariant time-separated correlators.

    Source: §I · p. 1, §IX · p. 10

  • Results

    The procedure was demonstrated in classical quantum-computer simulations of a two-plaquette D4 theory and a Z2 gauge field coupled to fermions.

    Source: §VIII · p. 8

  • Takeaways & Limitations

    The construction keeps physical and unphysical states unmixed under exact implementation and supports gauge-invariant correlators, including a temporal Wilson loop.

    Source: §IX · p. 10

    Demonstrations use perfect, error-free simulated hardware, and the gauge-subspace guarantee assumes the stated requirements are implemented exactly.

    Source: §VIII · p. 8, §IX · p. 10

Abstract

from arXiv · show

A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant operators is detailed. We demonstrate the procedure on small lattices, including the simulation of a 2+1D non-Abelian gauge theory.

I. INTRODUCTION

Classical obstacles to nonperturbative quantum-field-theory simulation and near-term hardware limits motivate quantum approaches, but gauge theories pose challenges across representation, preparation, evaluation, and renormalization. The paper proposes gauge-preserving Trotter evolution and discusses classical coupling matching.

I. INTRODUCTION

  • Bosonic gauge-field digitization is nontrivial; gauge fixing can restrict registers to physical states but increases evolution-circuit depth, and a practical non-Abelian method remains undescribed.Proposed representation approaches include finite subsets, Fock-state truncation, dual variables, optical representations, and prepotentials.Source: §I · p. 1
  • In strongly coupled theories, the dependence of asymptotic states on fundamental fields is unknown, while preparation methods chiefly target ground states.Proposals include variational methods, QPE, QAA, spectral combing, hybrid methods, and dimensional reduction; scattering simulations are hindered by state preparation.Source: §I · p. 1
  • Time-separated composite observables are harder to evaluate than single-time Hermitian observables, and real-time evolution lacks Euclidean imaginary-time state separation.The paper uses ancillary probe and control qubits for composite observables; state contamination may require new projection methods.Source: §I · p. 1
  • The paper maps a gauge-theory Hilbert space to a quantum computer and uses Trotterization to approximate propagation by U(t).The cited passage notes that such Trotterized evolution allows efficient simulation.Source: §I · p. 1
  • For arbitrary gauge theories with coupled matter, the proposed Trotterized evolution preserves gauge invariance without gauge fixing; digitization error is treatable, while noisy-gate effects remain hardware-dependent.The construction is connected to the transfer-matrix formalism [67].Source: §I · p. 1
  • A transfer-matrix mapping from Trotterization to a Euclidean action may let standard classical lattice methods compute renormalized couplings efficiently.This would match quantum-computer bare parameters to renormalized and physical ones.Source: §I · p. 2

II. PREREQUISITES

The prerequisites specify idealized qubit gates and group-valued registers as hardware-independent constructs. Continuous groups require approximate registers in finite qubit systems, but this work considers ideal registers and specifies additional gates for matter fields.

II. PREREQUISITES

  • The construction assumes idealized qubits and a sufficient universal gateset; algorithm correctness is independent of whether the primitive gates use superconducting qubits or trapped ions.The text gives Hadamard, T, and CNOT as examples of primitive gates.Source: §II · p. 2
  • A G-register represents a gauge-group element, with ideal Hilbert space CG and one basis state per group element.Multiple registers form a tensor-product Hilbert space.Source: §II · p. 2
  • Discrete groups admit finite ideal registers, whereas continuous Lie groups such as SU(3) require approximate finite-qubit registers with a qubit-count versus discretization-error tradeoff.The paper considers only ideal G-registers, leaving discretization-related complications unaddressed.Source: §II · p. 2
  • Gauge-register operations include inversion, multiplication, trace, and Fourier transform; the Fourier transform diagonalizes the kinetic part of the Trotterized evolution.Matter fields require additional multiplication and Fourier-transform gates, plus an inner-product gate.Source: §II · pp. 2–3
  • Scalar matter uses a truncated C^N-register, while fermionic matter needs no C^N-register or truncation.The C^N-register stores a component of C^N, with N the dimension of the gauge group's fundamental representation.Source: §II · p. 3

III. PURE GAUGE

The pure-gauge construction defines a Hamiltonian on the full gauge-theory space and Trotterizes it into implementable pieces. Because each term is gauge-invariant, Trotterization preserves gauge invariance at every order in Δt.

III. PURE GAUGE

  • The non-Abelian approach uses logical qubits and gates, is independent of quantum-computer geometry, and avoids ancillary G-registers.Similar approaches had been discussed previously [54] [59] [60] [61] [62] [77].Source: §III · p. 3
  • The gauge-field-operator, or position, basis simplifies plaquettes, Wilson lines, and matter-field inclusion, and more closely matches classical lattice-field-theory calculations.The propagation gate requirements are tabulated for lattices with N_P plaquettes and L links over time T in steps of Δt.Source: §III · p. 3, Table I · p. 4
  • Physical states are gauge-symmetrized states satisfying Gauss’s law, but this method does not require implementing the projection operator P.Initial states should lie within the physical subspace.Source: §III · pp. 3–4
  • The Hamiltonian acts on the full space H, with an implicitly defined physical-subspace Hamiltonian, and is motivated by recovering the Euclidean Wilson action.Its potential term sums spatial plaquettes, while the kinetic term describes free quantum-mechanical dynamics; the transfer-matrix formalism addresses the discrete-theory kinetic term.Source: §III · pp. 3–4
  • Each gauge-invariant Hamiltonian term can be Trotterized without breaking gauge invariance at any order in Δt.The construction is intended for quantum-computer implementation.Source: §III · p. 4
  • Local terms within each kinetic or potential part commute, so they can be applied individually and sequentially without approximation.The potential evolution is implemented using plaquette products in an ancillary register, while kinetic evolution diagonalizes via U_F before applying a diagonal unitary.Source: §III · p. 4

IV. THE LATTICE PATH INTEGRAL

The section connects Hamiltonian evolution to the Euclidean lattice path integral through a transfer matrix. This connection supports classical coupling determination and gives a route to Hamiltonians for discrete gauge groups.

IV. THE LATTICE PATH INTEGRAL

  • The transfer matrix approximates one unit of imaginary-time evolution and is constructed from kinetic and potential contributions using a split-operator approximation.The potential contribution resembles the spatial-plaquette product in the Wilson action; the kinetic term is related to timelike plaquettes.Source: §IV · pp. 4–5
  • Restricting the transfer matrix to physical states reproduces the usual lattice path integral with the Wilson action.The paper defines its transfer matrix on the full Hilbert space; projecting with P yields the usual physical-space construction.Source: §IV · pp. 4–5
  • The transfer-matrix mapping makes the bare couplings for desired renormalized physical parameters match those of the lattice action, enabling classical parameter determination.For example, appropriate bare parameters for a desired physical pion mass may be determined classically.Source: §IV · pp. 4–5
  • When the Trotter step equals the spatial lattice spacing, the evolution is an analytic continuation of isotropic lattice calculations, with Lorentz invariance recovered in the continuum limit.The kinetic Hamiltonian is recovered from the transfer matrix as the temporal lattice spacing tends to zero.Source: §IV · p. 5
  • For a discrete gauge group, the kinetic Hamiltonian can be obtained from the transfer-matrix contribution by taking its logarithm, provided that contribution is positive-definite.The construction uses H = -log TV - log TK.Source: §IV · p. 5

V. CORRELATORS

The section develops procedures for measuring time-separated correlators and temporally extended Wilson loops using unitary evolutions and perturbations. These procedures use finite differences and can, for some insertions, take the state outside the physical subspace.

V. CORRELATORS

  • An ancillary-qubit controlled-unitary procedure measures real and imaginary parts of unitary expectation values using σx and σy measurements, respectively.The ancilla is initialized in |+⟩.Source: §V · p. 6
  • For QCD, the Wilson plaquette correlator corresponds to a glueball propagator, and its nonunitary operator is accessed through derivatives of a unitary family generated by a time-dependent Hamiltonian perturbation.The construction uses the Hermiticity of Wµν(x).Source: §V · p. 6
  • Time-separated plaquette correlators are obtained from second derivatives of a perturbed-evolution expectation value, with finite differencing required in practice.A k-time correlator requires the kth numerical derivative.Source: §V · p. 6
  • The temporally extended Wilson-loop construction sums color indices to produce the gauge-invariant trace, while its perturbation is not gauge-invariant and takes the evolving state outside the physical subspace.The perturbation also requires diagonal phase gates on individual G-registers that are not specified among the earlier primitive gates.Source: §V · p. 6
  • For a correctly prepared initial state, the method gives zero for the non-gauge-invariant expectation value ⟨Tr Uik(t)Uij(0)⟩ when j ≠ k.Source: §V · p. 6

VI. SCALAR FIELDS

The section extends the gauge-field Hilbert space and evolution to include scalar matter fields. It specifies a five-part time step and notes that classical renormalization calculations can again be used.

VI. SCALAR FIELDS

  • Scalar matter transforming in a D-dimensional representation is added by tensoring D scalar degrees of freedom at each site with the gauge-field Hilbert space.The scalar site basis is expressed in occupation numbers.Source: §VI · pp. 6–7
  • The scalar and gauge fields transform so that φ†jUijφi is gauge-invariant, and the gauge-symmetrization operator acts on the expanded Hilbert space.Source: §VI · p. 7
  • A scalar-field time step applies plaquette, scalar hopping, scalar mass, gauge kinetic, and scalar kinetic gates in sequence.The mass and hopping gates commute because both are diagonal in the fiducial basis.Source: §VI · p. 7
  • The scalar-field extension can be connected through the transfer matrix to a classical Euclidean path integral, so renormalization calculations may again be performed classically.Source: §VI · p. 7
  • For a general Wilson line, the correlator construction requires a derivative at every timeslice; finite differencing makes this method implausible even in the far future.A more efficient method remains an open problem.Source: §VI · p. 7

VII. FERMION FIELDS

The section incorporates spinor matter using fermionic operators built from qubit-compatible bosonic operators. It addresses anticommutation and gauge transformation requirements, then constructs evolution from the fermionic Hamiltonian.

VII. FERMION FIELDS

  • The Jordan-Wigner transformation converts commuting qubit operators into anticommuting fermionic operators, at the cost of making local fermionic operators nonlocal in the bosonic representation.More efficient but more complex schemes for relating local fermionic operators to more local bosonic operators are noted.Source: §VII · p. 8
  • A single-fermion site register is exact and requires no approximation or truncation because its occupation is limited by the exclusion principle.Source: §VII · p. 8
  • In the presence of gauge fields, the bosonic operators used to construct fermions transform in the fundamental representation, allowing the fermionic operators to obey the required gauge transformation law.The Jordan-Wigner procedure itself needs no change because its σ matrices are independent of the gauge group.Source: §VII · p. 8
  • The spinor evolution is constructed similarly to the scalar-field evolution, with its kinetic and potential operators derived from the Dirac equation; a Z2-fermion implementation is demonstrated.Source: §VII · p. 8

VIII. DEMONSTRATION

The demonstrations apply the procedure to a two-plaquette D4 gauge theory and a Z2 gauge field coupled to fermions. Simulated plaquette and matter-density evolution are compared with exact results, and the D4 temporal Wilson loop is also evaluated.

D4 gauge theory

  • The D4 setup uses four gauge registers on the smallest two-dimensional lattice that cannot be reduced to a one-dimensional theory.Its Hamiltonian combines two spatial plaquette traces and the one-link kinetic term K.Source: §VIII · pp. 8–9, Figure 2 · p. 9
  • With Δt = 0.5, D4 plaquette errors become noticeable sooner at large times than with Δt = 0.2.The simulated calculation is noiseless, so its only systematic error is Trotterization.Source: §VIII · p. 9, Figure 3 · p. 9
  • The procedure is used to determine a D4 temporal Wilson loop as a function of loop time extent, with Trotterization and finite differencing as systematic errors.Each data point uses 2×10^5 measurements; the finite-difference simulations use Δt = 0.2.Source: Figure 4 · p. 9, §VIII · p. 9
  • Reducing the finite-difference deformation ε increases the required sample count, which scales as ε^-2.Smaller deformations are preferred for finite differencing.Source: §VIII · p. 9

Z2 gauge theory with fermions

  • In the Z2 staggered-fermion simulation, the leftmost-site density varies over time and differs from exact values through sampling error and Trotterization.The simulation uses 8 qubits, 36 CNOT gates per timestep, Δt = 0.6, and T = 30.Source: §VIII · p. 10

IX. DISCUSSION

The discussion presents a procedure for unitary evolution and time-separated correlators, with gauge-sector preservation conditional on exact implementation of the specified requirements. It reports demonstrations on two gauge theories and identifies practical implementation and noisy-gate behavior as open issues.

IX. DISCUSSION

  • The Trotterized evolution does not mix physical and unphysical states, provided the Section II requirements are implemented exactly.The procedure is intended for an idealized universal quantum computer.Source: §IX · p. 10
  • The procedure is demonstrated on two gauge theories, and the paper also discusses classical determination of bare couplings for quantum simulations.The demonstrations include a Z2 gauge theory with staggered fermions.Source: §IX · p. 10, Figure 5 · p. 10
  • Efficient implementation of the registers and gates remains open; noisy-gate effects on mixing and possible mitigation are left for future work.The paper leaves open whether mitigation is necessary.Source: §IX · p. 10

Appendix A: Implementation of a D4-register

Appendix A maps the D4 group register to three qubits and describes circuits for its group operations and Fourier transform. It also reports a search for an exact Fourier-transform circuit in the specified gate set.

Appendix A: Implementation of a D4-register

  • Three qubits encode D4 so the two least-significant bits specify rotation and the most-significant bit specifies whether to flip.The group is treated as a subgroup of U(2), with the two bits specifying rotation followed by a flip when the most-significant bit is 1.Source: Appendix A · p. 11
  • Inversion and multiplication use CNOT-only circuits, while the trace operation is a three-qubit controlled phase gate.The trace gate applies phases to |000⟩ and |010⟩ and no phase to the other states.Source: Appendix A · p. 11
  • A classical simulated-annealing search finds an exact Fourier-transform circuit within the stated Qiskit gate set.The text names Hadamard, CNOT, and π/8 gates; the Fourier transform diagonalizes the D4 momentum operator.Source: Appendix A · p. 11
  • The Fourier-transform circuit is shown for a D4 register, with least-significant qubits at the top.The accompanying figure identifies H as the Hadamard gate and begins to define T as a π… gate.Source: Figure 7 · p. 11

Appendix B: Positive-Definiteness of the Transfer Matrix

Appendix B proves positive-definiteness of the discrete-gauge-theory kinetic transfer matrix on the full Hilbert space for faithful representations. The proof reduces the claim to single-link operators and uses nonnegative terms in an order-by-order expansion in β.

Appendix B: Positive-Definiteness of the Transfer Matrix

  • For faithful representations, the kinetic transfer matrix is positive-definite on the full Hilbert space, not merely on the physical subspace.The appendix identifies the fundamental representation as an example of a faithful representation.Source: Appendix B · pp. 11–12
  • The full kinetic transfer matrix is a tensor product of single-link operators, so positivity follows when the single-link transfer matrix is positive-definite.The proof therefore reduces to establishing positive-definiteness for a single link.Source: Appendix B · p. 11
  • An order-by-order expansion in β expresses each term as nonnegative, using squared magnitudes of representation-matrix expressions.To rule out saturation at zero, the proof uses faithfulness of ρ and completeness of polynomials on G to infer that f(g) vanishes.Source: Appendix B · p. 12

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