Finite-Size Limits of Adaptive Pauli Prediction

Abstract: Can quantum data reliably answer questions chosen from earlier results? We prove that $M$ adaptive Pauli predictions on $n$ qubits require $Ω(\sqrt{\min{n,M}})$ copies at fixed accuracy. Published upper bounds make this tight for $M\le n$. Thus $n$ adaptive predictions cost $Θ(\sqrt n)$ copies, whereas a fixed list costs only $O(\log n)$. The lower bound uses diagonal states and commuting observables, identifying a statistical cost of adaptation. We also quantify how this cost increases with precision.