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。这些任务涵盖从状态区分和计量等基础问题，到量子概率近似正确（PAC）学习框架，以及最近提出的量子态层析成像的影子变体。然而，量子学习理论的众多方向迄今仍各自独立发展。我们提出一个通用的数学形式体系，用于描述通过经典-量子数据训练进行量子学习的过程，并检验所学假设对新数据的泛化能力。在此框架下，我们利用经典和量子信息论度量（衡量学习者假设对训练期间特定数据的依赖程度），证明了量子学习器期望泛化误差的上界。为此，我们借助量子最优输运和量子浓度不等式等工具，建立了去耦引理的非对易版本，这些引理是经典机器学习中近期信息论泛化界的基础。我们的框架涵盖了多种量子学习场景（如量子状态区分、量子态PAC学习、量子参数估计以及经典函数的量子PAC学习），并提供了直观可理解的泛化界。因此，我们的工作为量子学习的统一量子信息论视角奠定了基础。