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.

翻译：泛化是机器学习理论中的核心概念，然而对于量子模型而言，其分析主要依赖于基于模型整体容量而非具体学习函数的均匀上界。这些基于容量的均匀上界通常过于宽松，且对实际的训练和学习过程完全不敏感。先前的理论保证未能提供反映学习解具体性质（而非整个假设类的最坏情况行为）的非均匀、数据相关的上界。为解决这一局限，我们通过分析由量子通道（包括诸如电路中间测量和前馈等耗散操作）构成的层状电路，首次推导出了适用于一大类量子模型的PAC-Bayesian泛化上界。通过通道扰动分析，我们建立了依赖于学习参数矩阵范数的非均匀上界；我们将这些结果扩展到受对称性约束的等变量子模型；并通过数值实验验证了我们的理论框架。本工作提供了具有可操作性的模型设计见解，并为基础性工具奠定基础，从而更细致地理解量子机器学习中的泛化问题。