Concept learning of parameterized quantum models from limited measurements

TL;DR

This paper develops a unified theoretical framework for learning parameterized quantum models from limited measurements, quantifying the effects of sample size and measurement shots, and analyzing measurement noise impact.

Contribution

It introduces provable guarantees for learning quantum models that incorporate measurement shot effects and noise, advancing understanding of classical-quantum measurement interplay.

Findings

Increasing sample size improves learning performance.

Single-shot measurements are nearly as effective as multiple shots.

Measurement noise significantly affects classical surrogation of quantum models.

Abstract

Classical learning of the expectation values of observables for quantum states is a natural variant of learning quantum states or channels. While learning-theoretic frameworks establish the sample complexity and the number of measurement shots per sample required for learning such statistical quantities, the interplay between these two variables has not been adequately quantified before. In this work, we take the probabilistic nature of quantum measurements into account in classical modelling and discuss these quantities under a single unified learning framework. We provide provable guarantees for learning parameterized quantum models that also quantify the asymmetrical effects and interplay of the two variables on the performance of learning algorithms. These results show that while increasing the sample size enhances the learning performance of classical machines, even with single-shot estimates, the improvements from increasing measurements become asymptotically trivial beyond a constant factor. We further apply our framework and theoretical guarantees to study the impact of measurement noise on the classical surrogation of parameterized quantum circuit models. Our work provides new tools to analyse the operational influence of finite measurement noise in the classical learning of quantum systems.