Generalization in quantum machine learning from few training data

Journal Article · Mon Aug 22 00:00:00 EDT 2022 · Nature Communications

DOI:https://doi.org/10.1038/s41467-022-32550-3· OSTI ID:1882759

Caro, Matthias C.; Huang, Hsin-Yuan; Cerezo, M.; Sharma, Kunal; Sornborger, Andrew; Cincio, Lukasz; Coles, Patrick J.

Abstract

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 $$$$\sqrt{T/N}$$$$ $\sqrt{T/N}$ . When only K  ≪  T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to $$$$\sqrt{K/N}$$$$ $\sqrt{K/N}$ . 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.

• Research Organization: Los Alamos National Laboratory (LANL)
• Sponsoring Organization: German Academic Scholarship Foundation; J. Yang & Family Foundation; Technical University of Munich; USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1882759
• Alternate ID(s): OSTI ID: 1889979
• Report Number(s): LA-UR-21-31086; 4919; PII: 32550
• Journal Information: Nature Communications, Journal Name: Nature Communications Journal Issue: 1 Vol. 13; ISSN 2041-1723
• Publisher: Nature Publishing Group
• Country of Publication: United Kingdom
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
computer science
quantum information