Reporting the Accuracy of Approximate Quantum Circuit Simulation

Approximate simulators of quantum circuits routinely return outcome distributions and expectation values with no accompanying statement of how far those numbers may be from the exact result. The underlying error quantities are usually available inside the simulation and are simply discarded before the result reaches the user.

We describe ZCC-v0.1 and ZHF-v0.1, two small versioned protocols that attach a machine-checkable accuracy statement to every result: a measured single-run truncation bound for matrix product state simulation, an additive coefficient-mass bound for Pauli propagation, a convergence diagnostic where neither applies, and a direct-comparison fidelity for runs on quantum hardware.

We state precisely what each figure does and does not assert. In particular we give the condition under which the matrix product state bound is tight, and the worst case in which it is not, since this distinction is frequently elided in practice. We validate the bound against exactly computed answers, first over a structured sweep and then across 290 runs constructed specifically to falsify it. The per-outcome claim survives, with a factor of two in hand at worst, but two things do not: a natural tightening of the bound is shown to be unsound by explicit counterexample, and the derivation that motivates the bound is shown not to extend to deep sequential circuits, where the accumulated truncation weight understates the true infidelity by up to a factor of seven. We also show that a common way of assessing such bounds understates their tightness by two orders of magnitude by measuring the observable rather than the bound. We further argue that a simulation whose own error bound is vacuous should be refused rather than returned, and report the effect of implementing that rule. The mathematics is standard; the contribution is a specification, a reference implementation, a public verification surface, and the measurement that the reported figures actually hold.