Finite-Shot Operating Windows for Virtual Distillation and Symmetry Verification
Quantum error mitigation methods are usually compared by their infinite-shot bias, yet in an experiment the comparison is decided by a finite shot budget, by the instability of the estimator, and by the cost of each shot. We derive finite-shot mean-squared-error laws for virtual distillation (VD) and symmetry verification (SV) whose leading terms predict when one method overtakes the other and whose remainders are bounded with explicit constants. For VD the law contains the statistical bias and denominator instability of the quotient estimator, together with the sample size needed before the quotient can be trusted; for SV it separates the floor left by undetectable errors from the cost of postselection. A first-order decomposition of the noise generator shows which error components each method removes. Two such laws give a tie, a uniform dominance, or a trade-off whose crossing budget scales as $p^{-2}$ or $p^{-1}$ in the noise rate $p$; when the remainder bounds are sharp enough, they turn the leading crossing into a certified interval. In exact white-box QAOA simulations the predicted $p^{-2}$ scaling (fitted exponent $-1.97$) and the sign pattern of all pairwise comparisons ($300/300$ cells) are recovered, one pre-specified all-instance criterion is not met, and the crossings lie $3$--$13\%$ above the leading prediction; a certified interval containing the exact crossing is obtained only in a closed model. Gate-level simulations and runs on two IBM backends show why the ideal VD advantage disappears in the tested instances: interferometric overhead and denominator instability move its window beyond the tested budgets, where SV has the lower mean-squared error.