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 $\tilde{\mathcal{O}}\left(\frac{\hat{I}{\pm }^k}{\sqrt{N}}(k^{\frac{3}{2}}\sqrt{mn}+\sqrt{T\log T})\right)$ for $N$ training data points, $T$ trainable quantum gates, $n$ dimensional quantum circuit output, and $k$ bounded linear layers $\Vert F_i{\Vert }_F\le \hat{I}\pm$ where $i=1,\dots ,k$ and $F_i\in {\mathbb{R}}^{m\times n}$ interspersed with activation functions. This generalization bound decomposes into quantum and classical contributions, providing a theoretical framework to separate their influence and clarifying their interaction. 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.

Added activation function and multi layer formulations for the generalization bound