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arXiv:2504.08456 · Physical Review A

Generalization bounds in hybrid quantum-classical machine learning models

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

Published Dec 24, 2025• 2 citations
arXivHTMLPDF

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 O ̃ ( α k N ( k 3 2 m n + T log T ) ) for N training data points, T trainable quantum gates, n -dimensional quantum circuit output, and k bounded linear layers ∥ F i ∥ F ≤ α , where i = 1 , ⋯ , k and F ∈ R m × 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.

DOI
10.1103/6ckt-hlh8

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A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
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A large body of recent work has begun to explore the potential of parametrized quantum circuits PQCs as machine learning models, within the framework of hybrid quantum classical optimization. In particular, theoretical guarantees on the out of sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC based models that depend explicitly on the strategy used for data encoding. These imply bounds on the performance of trained PQC based models on unseen data. Moreover, our results facilitate the selection of optimal data encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC based models via trigonometric functions. Our generalization bounds emphasize the importance of well considered data encoding strategies for PQC based models
Quantum computing has emerged as one of the most influential paradigms capable of transforming computational processes that struggle under classical limitations. In recent years, hybrid quantum–classical algorithms have gained significant attention for their ability to leverage the strengths of both architectures, especially within the Noisy Intermediate-Scale Quantum (NISQ) era. This research paper explores a quantum-enhanced optimization framework designed to improve machine learning model performance through parameterized quantum circuits (PQCs). The goal of this study is to provide a practical, scalable approach that can be integrated into existing machine learning workflows without requiring fully developed fault-tolerant quantum hardware. The proposed framework uses classical preprocessing to encode data into quantum states, followed by quantum transformations that explore richer feature spaces due to quantum superposition and entanglement. A classical optimizer then updates the quantum circuit parameters based on measurement results, forming an end-to-end feedback loop. This hybrid model aims to demonstrate improved optimization efficiency, reduction in loss-function plateaus, and better generalization for specific ML tasks involving small to medium-sized datasets. This paper provides a comprehensive investigation of methodology, literature background, and experimental reasoning with statistical analysis, along with conceptual diagrams illustrating system architecture and circuit layouts. The outcomes further strengthen the viability of hybrid quantum–classical computation as a promising direction for advanced machine learning systems.
Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model's overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.