Certified Finite-Shot Operating Windows for Virtual Distillation and Symmetry Verification

Abstract

Quantum error mitigation methods are often compared through infinite-shot bias, but real experiments are decided by finite sampling budgets, estimator instabilities, and per-shot resource costs. We develop a certified finite-shot operating-window theory for comparing virtual distillation (VD) and symmetry verification (SV). For each method we prove a mean-squared-error law with explicit non-asymptotic remainders. For VD, the law exposes quotient-estimator bias, denominator-driven variance, and a concentration certificate for when the denominator is statistically resolved. For SV, it separates the residual bias from undetectable errors from the sampling penalty set by the acceptance probability. These laws induce a selection trichotomy: a two-method comparison is a tie, a uniform dominance relation, or a genuine tradeoff with a certified crossing window and a self-consistency test. Exact white-box experiments confirm the predicted $p^{-2}$ operating-window scale with fitted exponent $-1.97$ and show $300/300$ sign agreement in a pre-specified analysis; the single strict all-instance criterion not met is reported with its calibration analysis. Gate-level simulation and archived IBM hardware runs then test the windows under device conditions: idealized VD windows exist, but realistic interferometry overhead and denominator instability move them outside the tested resource range, while calibrated SV has lower MSE in the tested QAOA instances. The result is a regime statement, not a universal ranking: certified operating windows explain when mitigation advantages should appear or disappear, and keep coefficient-level validation separate from noisy-device evidence.