A PAC-Bayesian approach to generalization for quantum models

TL;DR

This paper develops the first PAC-Bayesian generalization bounds for quantum models, providing data-dependent insights that improve understanding of quantum machine learning's generalization capabilities.

Contribution

It introduces non-uniform, data-dependent PAC-Bayesian bounds for quantum models, extending to symmetry-constrained models and validating through numerical experiments.

Findings

Bounds depend on learned parameter norms

Applicable to layered quantum circuits with dissipative operations

Provides insights for quantum model design

Abstract

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.