A Validated Quantum-Circuit Testbed for the Lattice Schwinger Model

Abstract

Abstract. We present a validated computational testbed for studying product-formula errorin a lattice gauge theory. For the lattice Schwinger model (1+1D QED, Kogut–Susskindstaggered fermions, gauge field eliminated via Gauss’s law) we construct three 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 firsttwo are independent; the second and third share the Hamiltonian construction and Paulidecomposition, so their agreement verifies the gate decomposition and circuit implementationrather than the physics, which is validated against the exact solution instead. The gate-levelcircuit reproduces the classical Trotter propagator to 7.6×10−14 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 decompositionagainst direct matrix construction, in both cases to machine precision.We then demonstrate the testbed on three questions from the existing literature. First, thecommutator-scaling bound of Childs et al. (2021), evaluated with prefactors derived numerically rather than assumed, overestimates the measured operator-norm error by a factor between1.5 and 6.8 across nine parameter points (median 2.0), rising with system size and coupling;at the reference point the ratio is 1.74, constant to three significant figures over the testedrange ∆t ∈ [1/512, 1/16]. Second, operator-norm, state, and observable errors all give fittedexponents statistically consistent with p = 2 but markedly different prefactors; we introduce anormalisation-independent gap factor G to quantify this, finding G numerically constant overthe tested ∆t range and increasing across the four accessible system sizes. We show that thisgrowth reflects the growth of the state error while the local-observable error stays constantto five significant figures — the local-versus-global robustness established by Heyl, Hauke andZoller (2019), for which we claim no priority. Third, the ordering heuristics of Tranter et al.(2019), established on 44 molecular Hamiltonians, reproduce all four of their reported trendsat the parameter point tested here — the fourth in sharpened form, the norm-based estimatorranking orderings slightly backwards (ρ = −0.292) rather than merely failing to rank them.The contribution is the validated testbed and its convention bookkeeping rather than newtheory: all three demonstrations are measurements of known results, and in the case of theobservable gap, a re-derivation of a known result in a different language. Reference [5] studiesthe same model with the same decomposition and initial state, using a first-order formula inthe long-time limit; the present work uses the second-order symmetric formula at short timeand compares against the 2021 commutator-scaling bound, which postdates that study. Codefor all reported figures and tables is provided.