Generalization Bounds in Hybrid Quantum-Classical Machine Learning Models

Abstract: Hybrid classical-quantum models aim to harness the strengths of both quantum computing and classical machine learning, but their practical potential remains poorly understood. In this work, we develop a unified mathematical framework for analyzing generalization in hybrid models, offering insight into how these systems learn from data. We establish a novel generalization bound of the form $O\big( \sqrt{\frac{T\log{T}}{N}} + \frac{\alpha}{\sqrt{N}}\big)$ for $N$ training data points, $T$ trainable quantum gates, and bounded fully-connected layers $||F|| \leq \alpha$. This bound decomposes cleanly into quantum and classical contributions, extending prior work on both components and clarifying their interaction. We apply our results to the quantum-classical convolutional neural network (QCCNN), an architecture that integrates quantum convolutional layers with classical processing. Alongside the bound, we highlight conceptual limitations of applying classical statistical learning theory in the hybrid setting and suggest promising directions for future theoretical work.