A PAC-Bayesian approach to generalization for quantum models

2026-03-24

AI Summary

This work addresses the limitation of existing generalization analyses for quantum models, which predominantly rely on loose, data-independent uniform bounds that fail to capture data-dependent characteristics. By leveraging the PAC-Bayesian framework, this study establishes the first non-uniform, data-dependent generalization bound for layered quantum circuits incorporating dissipative operations—such as intermediate measurements and feedforward control. The analysis introduces a quantum channel perturbation approach that explicitly links the generalization bound to the norm of the learned parameter matrices. Furthermore, the framework is extended to equivariant quantum models under symmetry constraints. The resulting theoretical bounds are provably tighter than previous results, and numerical experiments corroborate their empirical validity, offering actionable theoretical guidance for the design and optimization of quantum machine learning models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• generalization
• quantum models
• PAC-Bayesian
• non-uniform bounds
• data-dependent bounds

Innovation

• PAC-Bayesian
• quantum generalization
• non-uniform bounds
• quantum channels
• equivariant quantum models