A Validated Three-Reference Quantum-Circuit Testbed for the Lattice Schwinger Model

Abstract

Abstract. We present a validated computational testbed for studying product-formula error in a lattice gauge theory. For the lattice Schwinger model (1+1D QED, Kogut–Susskind staggered fermions, gauge field eliminated via Gauss’s law) we construct three mutually independent references — exact diagonalization, classical second-order Trotter evolution, and a gate-level Qiskit circuit — and cross-check them against each other and against published conventions. The gate-level circuit reproduces the classical Trotter propagator to \(7.6 \times 10^{-14}\) after accumulated repetitions, so that the residual difference from exact evolution isolates the physical Trotter error cleanly; the Hamiltonian is verified term-by-term against Nguyen et al., and the Pauli decomposition against direct matrix construction, in both cases to machine precision. We then demonstrate the testbed on three questions from the existing literature. First, the commutator-scaling bound of Childs et al. (2021), evaluated with prefactors derived numerically rather than assumed, overestimates the measured operator-norm error by a factor 1.74, constant across five decades of \(\Delta t\). Second, operator-norm, state, and observable errors all give fitted exponents statistically consistent with \(p = 2\) but markedly different prefactors; we introduce a normalisation-independent gap factor \(G\) to quantify this, finding \(G\) numerically constant over the tested \(\Delta t\) range and increasing across the four accessible system sizes. Third, the ordering heuristics of Tranter et al. (2019), established on 44 molecular Hamiltonians, reproduce all four of their reported trends at the parameter point tested here. The contribution is the validated testbed and its convention bookkeeping rather than new theory: all three demonstrations are measurements of known results in a Hamiltonian class where they had not previously been checked. Code for all reported figures and tables is provided.