Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model

Technical Summary: Symmetry-Preserving and Scalable Variational Circuits for the Lattice Gross-Neveu Model

Problem Statement
The paper addresses the challenge of preparing ground states and sector-resolved excited states for the (1+1)-dimensional lattice Gross-Neveu model, a theory of $N_f$ relativistic fermion flavors coupled by a local four-fermion interaction. While tensor-network methods (e.g., Matrix Product States, DMRG) are highly effective for 1D systems, quantum simulation offers a direct route to unitary time evolution and scattering calculations. However, existing variational quantum eigensolver (VQE) proposals often utilize hardware-efficient ansätze that violate the Hamiltonian's symmetries, specifically the conservation of total fermion number ($N$) and individual flavor numbers ($N_b$). The authors aim to construct variational circuits that strictly preserve these symmetries to improve preparation accuracy and develop a scalable adaptive approach (SC-ADAPT-VQE) that maintains these constraints while addressing the problem of circuit architecture selection.

Methodology
The authors define the lattice Hamiltonian using staggered fermions with open boundary conditions. They propose two distinct approaches for state preparation:

1. Symmetry-Preserving Fixed Variational Ansätze:

   • Reference State: Both families start from a computational basis state with odd sites filled, satisfying specific charge sectors.
   • Total-Number Conserving (TNC): Uses "B" blocks that allow occupation transfer between flavors while preserving the total number $N$.
   • Flavor-Number Conserving (FNC): Uses "B" blocks for same-flavor pairs and "D" blocks for cross-flavor pairs. The D blocks introduce cross-flavor correlations without transferring occupation, thereby strictly conserving every individual flavor number $N_b$.
   • Structure: Both families share the same circuit graph, layer organization, and local-phase parametrization. The FNC family imposes a stronger constraint by removing cross-flavor exchange parameters.

2. SC-ADAPT-VQE (Scalable Circuit-ADAPT-VQE):

   • Pool Construction: The authors construct a problem-specific pool of generators derived from the Gross-Neveu Hamiltonian. This includes:
     • Mass-hopping volume and surface generators.
     • Interaction-hopping generators, specifically a cross-flavor family derived from the commutator of the interaction and hopping terms.
   • Scalability: The pool is organized into translated bulk sums and boundary-sensitive surface corrections. For a fixed flavor number and range caps, the number of shared variational parameters remains independent of volume, while the pre-compilation local-factor count grows linearly with volume.
   • Selection: Operators are selected based on energy gradients ($g_{FD}$) and reoptimized after each addition.

Key Contributions

• Symmetry-Enforcing Architectures: The construction of fixed ansätze that strictly enforce flavor-number conservation, demonstrating that restricting the variational search to the correct symmetry sector improves energy accuracy compared to total-number-conserving or non-symmetry-preserving approaches.
• SC-ADAPT Pool for Gross-Neveu: The development of a compact, range-truncated operator pool that includes interaction-sensitive cross-flavor generators, ensuring all selected operators commute with flavor charges.
• Finite-Volume Transfer: A demonstration that a fixed sequence of generators and optimized parameters from a smaller volume can be transferred to larger volumes without re-optimization or operator selection, maintaining consistent energy errors.

Results
The authors benchmarked their methods using classical simulations (statevector and Matrix Product States) against exact diagonalization and DMRG references.

• Fixed Ansatz Benchmarks:

  • For massive systems with $N_f = 1$ to $4$ and up to 30 qubits, the flavor-number-conserving ansatz achieved relative energy errors of approximately $1 \times 10^{-5}$ to $2 \times 10^{-4}$.
  • In two-flavor comparisons, the flavor-number-conserving ansatz consistently outperformed the total-number-conserving ansatz, with errors roughly an order of magnitude smaller in some cases.
  • These errors were smaller than those reported in previous literature (Ref. [17]) for comparable system sizes, though the authors note differences in discretization (staggered vs. standard Dirac) and resources prevent a direct algorithmic ranking.

• SC-ADAPT Benchmarks:

  • For $N_f = 1, 2, 3$ and up to 30 qubits, the SC-ADAPT approach achieved relative energy errors below $2.1 \times 10^{-3}$.
  • The operator selection sequence showed stability across volumes for the one-flavor case (LD=12-15), where a frozen 10-generator sequence from LD=12 yielded errors of $1.97 \times 10^{-3}$ to $2.03 \times 10^{-3}$ at LD=13-15 without re-optimization.

• Large-Scale Transfer (One-Flavor):

  • Using Matrix Product State (MPS) evaluations, the one-flavor circuits were extended to 60–400 qubits (LD=30 to 200).
  • With frozen parameters from LD=12 (and an extrapolation prescription for LD=200), the energy deviations from DMRG references ranged from 0.35% to 1.76% at a bond dimension of 256. The authors attribute these deviations partly to finite-bond MPS simulation errors.

• Excited States:

  • Using sequential deflation within a fixed flavor-number sector, the authors prepared the lowest four states in the $(N_0, N_1) = (2, 2)$ sector for 8-qubit systems.
  • The method successfully distinguished states within the conserved sector from the global spectrum. For degenerate subspaces (e.g., the massive rank-3 target), the authors utilized basis-independent projector fidelities to assess performance, achieving near-unity overlaps.

Significance and Claims
The paper claims that incorporating exact charge conservation (specifically flavor-number conservation) into variational circuits provides a practical benefit, yielding lower energy errors than less constrained approaches. The SC-ADAPT construction is presented as a scalable method where the shared parameter count is volume-independent, addressing the "curse of dimensionality" in variational parameter scaling.

The authors are modest regarding the scalability claims:

• They state that while the pre-compilation factor count is linear in volume, "fixed-accuracy performance at arbitrary volumes remains unestablished."
• The large-volume transfer tests were limited to the one-flavor case; for two flavors, the stabilization of the full operator sequence was not yet established.
• The results are specific to finite lattices at fixed spacing; continuum extrapolation and renormalized mass-gap scaling are identified as separate objectives not addressed in this work.
• The demonstrated accuracy for large volumes relies on classical simulations (MPS/Statevector) and does not yet account for hardware noise or compilation overheads.

In conclusion, the work establishes a framework for symmetry-preserving variational circuits in the Gross-Neveu model, demonstrating improved ground-state preparation accuracy and a pathway for scalable circuit construction, while acknowledging the remaining challenges in achieving fixed accuracy at arbitrary volumes and extending these results to noisy hardware.