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.

翻译：经典学习量子态可观测量期望值是学习量子态或量子通道的一种自然变体。虽然学习理论框架已确立了学习此类统计量所需的样本复杂度及每个样本的测量次数，但这两个变量之间的相互作用此前尚未得到充分量化。在本工作中，我们将量子测量的概率特性纳入经典建模考量，并在统一的学习框架下讨论这些量。我们为学习参数化量子模型提供了可证明的保证，这些保证同时量化了两个变量对学习算法性能的不对称影响及相互作用。结果表明：虽然增加样本量能提升经典机器的学习性能（即使使用单次测量估计），但增加测量次数带来的改进在超过常数因子后渐近趋于微不足道。我们进一步应用所提框架及理论保证，研究了测量噪声对参数化量子电路模型经典代理的影响。本工作为分析有限测量噪声在量子系统经典学习中的操作影响提供了新的分析工具。