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.

本研究解决了在量子状态或通道学习中，量子测量的概率性质与样本复杂性之间相互影响的不足量化问题。提出了一种统一的学习框架，并给出了学习参数化量子模型的可证明保障，揭示了测量噪声对经典学习的影响，为分析量子系统中的有限测量噪声提供了新的工具。

从有限测量中学习参数化量子模型的概念