Recent years have seen significant activity on the problem of using data for the purpose of learning properties of quantum systems or of processing classical or quantum data via quantum computing. As in classical learning, quantum learning problems involve settings in which the mechanism generating the data is unknown, and the main goal of a learning algorithm is to ensure satisfactory accuracy levels when only given access to data and, possibly, side information such as expert knowledge. This article reviews the complexity of quantum learning using information-theoretic techniques by focusing on data complexity, copy complexity, and model complexity. Copy complexity arises from the destructive nature of quantum measurements, which irreversibly alter the state to be processed, limiting the information that can be extracted about quantum data. For example, in a quantum system, unlike in classical machine learning, it is generally not possible to evaluate the training loss simultaneously on multiple hypotheses using the same quantum data. To make the paper self-contained and approachable by different research communities, we provide extensive background material on classical results from statistical learning theory, as well as on the distinguishability of quantum states. Throughout, we highlight the differences between quantum and classical learning by addressing both supervised and unsupervised learning, and we provide extensive pointers to the literature.

翻译：近年来，利用数据学习量子系统特性或通过量子计算处理经典/量子数据的问题引发了大量研究活动。与经典学习类似，量子学习问题涉及数据生成机制未知的场景，学习算法的主要目标是在仅能访问数据及可能存在的专家知识等辅助信息时，确保令人满意的精度水平。本文通过聚焦数据复杂度、副本复杂度和模型复杂度，运用信息论技术系统回顾了量子学习的复杂度问题。副本复杂度源于量子测量的破坏性——这种测量会不可逆地改变待处理量子态，从而限制可从量子数据中提取的信息量。例如，与经典机器学习不同，在量子系统中通常无法使用同一份量子数据同时评估多个假设的训练损失。为使本文具备自洽性并便于不同研究群体理解，我们提供了关于统计学习理论经典成果及量子态区分性的详尽背景资料。全文通过监督学习与无监督学习两个维度，突出量子学习与经典学习的差异，并提供了丰富的文献索引。