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Generalization Bounds in Hybrid Quantum-Classical Machine Learning Models

Tongyan Wu, Amine Bentellis, Alona Sakhnenko, Jeanette Miriam Lorenz

TL;DR

This work addresses the theoretical question of how generalization scales in hybrid quantum-classical machine learning models. It develops a unified framework using covering numbers and Dudley’s entropy integral to bound the generalization error of hybrid QMLMs and derives a decomposed bound that separates quantum and classical contributions. The main result is a bound of the form $\tilde{O}\left(\frac{α^{k}}{\sqrt{N}}\left(k^{3/2}\sqrt{m n}+\sqrt{T\log T}\right)\right)$, highlighting how the number of trainable quantum gates $T$, the classical network depth $k$, and the Frobenius-norm bound $α$ interact with data size $N$ and output dimension $n$. This provides theoretical guidance for designing hybrid architectures that balance quantum circuit depth and classical expressivity while preserving strong generalization, though it also notes limitations in current bounds and the need for empirical validation to guide optimal quantum/classical resource allocation.

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 $\tilde{\mathcal O}\left( \tfrac{α^{k}}{\sqrt{N}}\, \big( k^{\tfrac{3}{2}}\sqrt{m n}\;+\;\sqrt{T\log T}\big) \right)$ 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.

Generalization Bounds in Hybrid Quantum-Classical Machine Learning Models

TL;DR

This work addresses the theoretical question of how generalization scales in hybrid quantum-classical machine learning models. It develops a unified framework using covering numbers and Dudley’s entropy integral to bound the generalization error of hybrid QMLMs and derives a decomposed bound that separates quantum and classical contributions. The main result is a bound of the form , highlighting how the number of trainable quantum gates , the classical network depth , and the Frobenius-norm bound interact with data size and output dimension . This provides theoretical guidance for designing hybrid architectures that balance quantum circuit depth and classical expressivity while preserving strong generalization, though it also notes limitations in current bounds and the need for empirical validation to guide optimal quantum/classical resource allocation.

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 for training data points, trainable quantum gates, dimensional quantum circuit output, and bounded linear layers where and 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.

Paper Structure

This paper contains 15 sections, 16 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Generalization
  5. Complexity measures
  6. Rademacher complexity
  7. Covering Number
  8. Results
  9. Discussion
  10. Methods
  11. Classical Part: Covering Numbers in Neural Networks
  12. Quantum Part: Generalization Bound in QMLMs
  13. Generalization Bound in the Hybrid QMLM
  14. Conclusion
  15. Acknowledgements

Key Result

Theorem 1

Consider arbitrary spaces $\mathcal{X}, \mathcal{Y}$. We define $\mathcal{Z}:=\mathcal{X} \times \mathcal{Y}$ and a real-valued hypothesis class $H$ on $\mathcal{Z}$. For any $\delta>0$ and any probability measure $\mathbb{P}$ on $\mathcal{Z}$ we have with probability at least $(1-\delta)$ for the t where $c > 0$ is some constant and $\ell\circ H:=\{(x, y) \mapsto \ell(h, y, x) \mid h \in H\}$ is

Figures (1)

  • Figure 1: Visualization of the proof structure: We present metric entropy/covering number bounds for both quantum and classical components, demonstrating that the hybrid metric entropy bound can be expressed through both. We then utilize this framework to derive our generalization bound via Dudley's entropy integral and Rademacher bounds.

Theorems & Definitions (25)

  • Definition 1: Empirical Rademacher Complexity
  • Theorem 1: Rademacher Generalization BoundsABartlett1999
  • Definition 2
  • Theorem 2: Dudley's Entropy Integral Shalev2014
  • Lemma 1: Metric Entropy for classical $k$-Layer Neural Networks
  • Theorem 3: Generalization Bound for $k$-layer Neural Networks
  • Lemma 2
  • Theorem 4: Generalization Bound for QMLM Caro2022
  • Theorem 5: Generalization Bound for Repeated Local Gates Caro2022
  • Theorem 6: Generalization Bound in Quantum-Classical Hybrid Models
  • ...and 15 more