Empirical Risk Minimization and Uniform Convergence for Probabilistically Observed and Quantum Measurement Hypothesis Classes

We continue the study of the learnability of quantum measurement classes in the setting where the learner is given access only to prepared quantum states, aiming for necessary and sufficient conditions for PAC learnability, along with corresponding sample complexity bounds. In the quantum setting, in contrast with the classical probabilistically observed case, sampled states are perturbed when a quantum measurement is applied, according to the Born rule, so that distinct samples in the training data cannot be arbitrarily reused. We first probe the results from previous works on this setting. We show that the empirical risk defined in previous works and matching the definition in the classical theory can fail to satisfy the uniform convergence property enjoyed in the classical learning setting for classes that we can show to be PAC learnable. Moreover, we show that VC dimension generalization upper bounds in previous work are in many cases infinite, even for measurement classes defined on a finite-dimensional Hilbert space. We then show that, nonetheless, every measurement class defined on a finite-dimensional Hilbert space is PAC learnable via a modification of the ERM rule.