# Structural risk minimization for quantum linear classifiers

arXiv: https://arxiv.org/abs/2105.05566 · HTML: https://mlsanity.com/paper/2105.05566/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 41 references that are in the map.
- First posted on arXiv 12 May 2021.

## Builds on (most cited first, 12 of 12)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [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)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Pseudo-dimension of quantum circuits](https://mlsanity.com/paper/2002.01490/) (2020)
- [On the statistical complexity of quantum circuits](https://mlsanity.com/paper/2101.06154/) (2021)
- [Practical application improvement to Quantum SVM: theory to practice](https://mlsanity.com/paper/2012.07725/) (2020)
- [Generalization in Quantum Machine Learning: a Quantum Information Perspective](https://mlsanity.com/paper/2102.08991/) (2021)

## Built on by (most cited first, 24 of 24)

- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Shadows of quantum machine learning](https://mlsanity.com/paper/2306.00061/) (2023)
- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix](https://mlsanity.com/paper/2212.04471/) (2022)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [On the expressivity of embedding quantum kernels](https://mlsanity.com/paper/2309.14419/) (2023)
- [Dynamical simulation via quantum machine learning with provable generalization](https://mlsanity.com/paper/2204.10269/) (2022)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Statistical Complexity of Quantum Learning](https://mlsanity.com/paper/2309.11617/) (2023)
- [Overfitting in quantum machine learning and entangling dropout](https://mlsanity.com/paper/2205.11446/) (2022)
- [Double descent in quantum kernel methods](https://mlsanity.com/paper/2501.10077/) (2025)
- [Coreset selection can accelerate quantum machine learning models with provable generalization](https://mlsanity.com/paper/2309.10441/) (2023)
- [Stability and Generalization of Quantum Neural Networks](https://mlsanity.com/paper/2501.12737/) (2025)
- [Kernel-based optimization of measurement operators for quantum reservoir computers](https://mlsanity.com/paper/2602.14677/) (2026)
- [On the Generalization of Adversarially Trained Quantum Classifiers](https://mlsanity.com/paper/2504.17690/) (2025)
- [Quantum Circuit-Based Learning Models: Bridging Quantum Computing and Machine Learning](https://mlsanity.com/paper/2602.00048/) (2026)
- [Hybrid Classical-Quantum architecture for vectorised image classification of hand-written sketches](https://mlsanity.com/paper/2407.06416/) (2024)
- [Multitask Learning for Earth Observation Data Classification with Hybrid Quantum Network](https://mlsanity.com/paper/2601.22195/) (2026)
- [A Theory of Finite-Noise Optima and Generalization in Quantum Machine Learning](https://mlsanity.com/paper/2608.24229/) (2026)

## Abstract (arXiv)

Quantum machine learning (QML) models based on parameterized quantum circuits are often highlighted as candidates for quantum computing's near-term ``killer application''. However, the understanding of the empirical and generalization performance of these models is still in its infancy. In this paper we study how to balance between training accuracy and generalization performance (also called structural risk minimization) for two prominent QML models introduced by Havl'ivcek et al. (Nature, 2019), and Schuld and Killoran (PRL, 2019). Firstly, using relationships to well understood classical models, we prove that two model parameters -- i.e., the dimension of the sum of the images and the Frobenius norm of the observables used by the model -- closely control the models' complexity and therefore its generalization performance. Secondly, using ideas inspired by process tomography, we prove that these model parameters also closely control the models' ability to capture correlations in sets of training examples. In summary, our results give rise to new options for structural risk minimization for QML models.