Reliable Adaptive Stopping for Krylov-Shadow Quantum Fisher Information Estimation

Scalable quantum Fisher information (QFI) estimation becomes actionable when the numerical estimate is paired with a trustworthy stopping decision. Krylov-shadow QFI estimation has two resource directions: the Krylov order sets the population resolution, whereas the sample count controls statistical uncertainty at that resolution. We show that treating these directions as one can produce false stops, where a width-based rule reports a narrow interval around a biased low-order estimate. We turn adaptive stopping into a two-component reliability problem, separating Krylov truncation from finite-sample uncertainty, and introduce AKS-QFI, a component-aware stopping interface for Krylov-shadow estimators. On a noisy mixed-state benchmark at $n=4$ qubits, width-only stopping has false-stop rates from $0.16$ to $0.68$. Under the same resource limit, AKS-QFI returns no false success declarations; after recalibrating Krylov resolution and sample counts, it returns accurate success declarations at true 5% relative tolerance. These results make adaptive stopping a reliability layer for shadow-based QFI estimation.