Generalization in quantum machine learning from few training data

By Matthias C. Caro, Hsin-Yuan Huang, M. Cerezo, Andrew T. Sornborger, Lukasz Cincio, Patrick J. Coles. Nature Communications, 2021

559 citations recorded by the source corpus.

Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number N of training data points. We show that the generalization error of a quantum machine learning model with T trainable gates scales at worst as T/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{T/N}$$\end{document}. When only K ≪ T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to K/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{K/N}$$\end{document}. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data. The power of quantum machine learning algorithms based on parametrised quantum circuits are still not fully understood. Here, the authors report rigorous bounds on the generalisation error in variational QML, confirming how known implementable models generalize well from an efficient amount of training data.

Subfields: Artificial Intelligence

View Paper: https://www.semanticscholar.org/paper/297a6890413050cca93e1897a318b0ffe9e520b3

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Citations in the corpus

Quantum variational algorithms are swamped with traps (2022, unknown): “As a second example, this time in the unitary compiling setting of qUSQ, we can evaluate the commonly used procedure of measuring the inner product or average fidelity of n-qubit states between a target unitary U∗ and a variationally chosen unitary V (θ) using statistical queries analogous to a swap”
Challenges and opportunities in quantum machine learning (2022, background): “Recent theoretical analysis of QNNs shows that their prediction performance is closely linked to the number of independent parameters in the QNN, with good generalization obtained when the amount of training data is roughly equal to the number of parameters [35].”