An Information Bottleneck for Learning Quantum Channels via Coherent Information

Generalization theory for learning from quantum data has recently been unified under an information theoretic lens with bounds controlling the generalization gap via the mutual information between the hypothesis of a learner and its training data. However, extension of such a framework to dynamic quantum channels plausibly the more practically relevant learning target. Given the ubiquity of parameterized quantum circuits as hypothesis classes for unknown unitary dynamics, it has remained an open problem, and existing bounds for this setting rely on abstract quantum mutual information terms that cannot be measured or actively controlled during training. In this paper, we introduce a framework, Coherently Regularized Quantum Channel learning that replaces this unmeasurable quantity with the coherent information of the hypothesis channel, a defined estimable capacity metric. Using a decoupled reference state construction and quantum optimal transport techniques, and Petz's variational for quantum relative entropy, we prove a high probability generation bound. These bounds confirm that the true risk of a learned hypothesis channel is controlled by its empirical risk and a scaling with its maximum coherent information. Maximum coherent information is measurable, resulting a training time regularizer that acts as an information bottleneck, perturbing hypothesis channels from encoding spurious, query-specific quantum correlations. We validate our approach for learning unknown random Clifford circuits under a restricted query budget, which showcases that coherently regularized training yields a smaller generalization gap than unregularized empirical risk minimization as a function of the gap tracking, and that shallow target circuits are learnable from substantially fewer queries.