Universal Sample Complexity Bounds in Quantum Learning Theory via Fisher Information matrix

In this work, we show that the sample complexity (equivalently, the number of measurements) required in quantum learning theory within a general parametric framework, is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error and with high success probability under maximum likelihood estimation. The upper bound is governed by the supremum of the largest diagonal entry of the inverse Fisher information matrix, while the lower bound is characterized by any diagonal element evaluated at arbitrary parameter values. We then apply the general bounds to Pauli channel learning and to the estimation of Pauli expectation values in the asymptotic small-error regime, and recover the previously established sample complexity through considerably streamlined derivations. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value estimation without quantum memory. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators. We derive the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli expectation estimation with entangled probes. Finally, we highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, learning sample complexity is governed by the inverse Fisher information matrix.