Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.

翻译：学习任务在量子信息与计算中扮演着日益重要的角色。其范围涵盖从态分辨和计量学等基本问题，到量子概率近似正确（PAC）学习框架，乃至最近提出的态层析的“影子”变体。然而，量子学习理论的诸多方向迄今仍各自独立发展。我们提出了一个描述量子学习的通用数学形式：通过对经典-量子数据进行训练，然后测试所学假设对新数据的泛化能力。在此框架下，我们证明了量子学习器期望泛化误差的界，其表达形式为经典与量子信息论量，这些量用于度量学习器的假设对训练期间所见特定数据的依赖程度。为实现这一目标，我们利用量子最优传输和量子集中不等式等工具，建立了非对易版本的解耦引理，这些引理构成了近期经典机器学习信息论泛化界的基础。我们的框架涵盖并为多种量子学习场景提供了直观易懂的泛化界，例如量子态分辨、量子态PAC学习、量子参数估计以及量子方式PAC学习经典函数。由此，我们的工作为建立统一的量子信息论视角来理解量子学习奠定了基础。