A PAC-Bayesian approach to generalization for quantum models
This work addresses the need for more precise generalization analysis in quantum machine learning, providing foundational tools for model design, though it is incremental as it adapts classical PAC-Bayesian theory to quantum contexts.
The authors tackled the problem of loose generalization bounds in quantum machine learning by deriving the first PAC-Bayesian bounds for quantum models, resulting in non-uniform, data-dependent bounds that depend on learned parameter norms and are validated numerically.
Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model's overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.