# Generalization Bounds in Hybrid Quantum-Classical Machine Learning Models

arXiv: https://arxiv.org/abs/2504.08456 · HTML: https://mlsanity.com/paper/2504.08456/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 30 references that are in the map.
- First posted on arXiv 11 Apr 2025.

## Builds on (most cited first, 22 of 22)

- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [On the Quantum versus Classical Learnability of Discrete Distributions](https://mlsanity.com/paper/2007.14451/) (2020)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Recent advances in deep learning theory](https://mlsanity.com/paper/2012.10931/) (2020)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [Generalization Bounds for Convolutional Neural Networks](https://mlsanity.com/paper/1910.01487/) (2019)
- [Do Quantum Circuit Born Machines Generalize?](https://mlsanity.com/paper/2207.13645/) (2022)
- [Quantum-classical convolutional neural networks in radiological image classification](https://mlsanity.com/paper/2204.12390/) (2022)
- [Effective dimension of machine learning models](https://mlsanity.com/paper/2112.04807/) (2021)
- [The Quantum Path Kernel: a Generalized Quantum Neural Tangent Kernel for Deep Quantum Machine Learning](https://mlsanity.com/paper/2212.11826/) (2022)
- [A hybrid quantum-classical approach for inference on restricted Boltzmann machines](https://mlsanity.com/paper/2304.12418/) (2023)

## Built on by (most cited first, 3 of 3)

- [A PAC-Bayesian approach to generalization for quantum models](https://mlsanity.com/paper/2603.22964/) (2026)
- [Is data-efficient learning feasible with quantum models?](https://mlsanity.com/paper/2508.19437/) (2025)
- [Q-DIVER: Integrated Quantum Transfer Learning and Differentiable Quantum Architecture Search with EEG Data](https://mlsanity.com/paper/2603.28122/) (2026)

## Abstract (arXiv)

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 tildemathcal Oleft( tfracαk√N, big( k^tfrac32√m n;+;√Tlog Tbig) right) 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, dots, k  and Fi ∈ 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.