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  • Open Access

Generalization bounds in hybrid quantum-classical machine learning models

Tongyan Wu1,*, Amine Bentellis2, Alona Sakhnenko1, and Jeanette Miriam Lorenz3,†

  • *Contact author: tongyan.wu@tum.de
  • †Contact author: jeanette.miriam.lorenz@iks.fraunhofer.de
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Phys. Rev. A 113, 022425 – Published 17 February, 2026

DOI: https://doi.org/10.1103/6ckt-hlh8

Abstract

Hybrid quantum-classical 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 generalization bound of the form Õ(αkN(k32mn+TlogT)) for N training data points, T trainable quantum gates, n-dimensional quantum circuit output, and k bounded linear layers ∥Fi∥F≤α, where i=1,⋯,k and F∈Rm×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.

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Phys. Rev. A 113, 022425– Published 17 February, 2026
  • Received 26 May 2025
  • Accepted 24 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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