Sampling and Krylov-Truncation Uncertainty in Quantum Fisher Information Estimation

Estimating quantum Fisher information (QFI) from randomized measurements with a truncated Krylov representation introduces two errors: sampling error around the fixed-order target and truncation error relative to the full-space QFI. Bootstrap intervals quantify the former but can become narrow around a biased target, making width-based stopping unreliable. We formulate adaptive stopping as selective uncertainty quantification, with separate sampling and truncation radii and abstention when their sum does not support the requested tolerance. For independent confirmation data, we prove finite-horizon false-declaration control provided that the combined radius is a valid total-error bound. We also introduce Krylov-first positivity stabilization to mitigate finite-sample plug-in bias. With frozen within-instance truncation calibration, a six-qubit experiment using \(\lvert+\rangle^\otimes6\) at \(K=48<64\) yields 19 declarations in 20 held-out runs, no observed false declarations, and 4.04% median relative error at a 10% target. Under matched random draws, full-space positivity produces 20 false declarations, whereas Krylov-first stabilization produces 16 correct declarations and four abstentions. The results demonstrate accurate selective QFI decisions at non-full Krylov resolution while identifying transferable truncation control as the central scalability challenge.