Generalization in quantum machine learning from few training data

TL;DR

This paper analyzes the generalization performance of quantum machine learning models trained on limited data, providing bounds that suggest efficient training and promising applications in quantum error correction and phase transition classification.

Contribution

It derives theoretical bounds on the generalization error in QML, showing that good generalization can be achieved with few training data points, especially when only a subset of gates change significantly.

Findings

Generalization error scales as rac{rac{T}{N}}

Error improves to rac{rac{K}{N}} when only K gates change

Quantum state classification across phase transitions requires minimal training data

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