Generalization Bounds in Hybrid Quantum-Classical Machine Learning Models
This work addresses the theoretical gap in hybrid quantum-classical models for researchers in quantum machine learning, though it is incremental as it builds on existing statistical learning theory.
The authors tackled the problem of understanding generalization in hybrid quantum-classical machine learning models by developing a unified mathematical framework, resulting in a novel generalization bound that decomposes into quantum and classical contributions.
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 $\tilde{\mathcal O}\left( \tfrac{α^{k}}{\sqrt{N}}\, \big( k^{\tfrac{3}{2}}\sqrt{m n}\;+\;\sqrt{T\log T}\big) \right)$ for $N$ training data points, $T$ trainable quantum gates, $n$ dimensional quantum circuit output, and $k$ bounded linear layers $ \|F_i\|_F \leq α$ 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.