Global Minimax Limits for Quantum Estimation with Oracle Access: The Case of Amplitude Estimation

Abstract We formulate quantum amplitude estimation as a quantum statistical decision problem and study its fundamental limits under oracle access. For the mean squared error loss, we establish a global, non-asymptotic minimax lower bound, showing that any quantum estimation procedure making at most oracle queries—allowing adaptive strategies, biased estimators, and entanglement-assisted measurements—must incur a worst-case risk of order , uniformly over compact subsets of the parameter space. The lower bound is obtained by viewing oracle access as a quantum statistical experiment and applying a classical two-point minimax reduction in the sense of Le Cam 8-10, based on oracle indistinguishability bounds derived from hybrid arguments. This approach yields global risk guarantees that are not captured by local or quantum Fisher-information-based analyses 11-13. We further show that iterative quantum amplitude estimation (IQAE) achieves the same risk scaling under the same oracle model and loss function. As a consequence, IQAE is minimax optimal among all quantum procedures with oracle access. Our results provide a complete decision-theoretic characterization of quantum amplitude estimation and certify the optimality of IQAE in a strict global minimax sense.