Near-optimal Prediction Error Estimation for Quantum Machine Learning Models

TL;DR

This paper establishes tight bounds on the prediction error of quantum machine learning models based on training data size and model complexity, supported by numerical simulations.

Contribution

It provides the first tight prediction error bounds for QML models trained on finite data, linking error to the number of gates and training set size.

Findings

Derived covering and packing number bounds for QML models.

Established tight prediction error bounds in terms of model and data parameters.

Numerical simulations support the theoretical bounds.

Abstract

Understanding the theoretical capabilities and limitations of quantum machine learning (QML) models to solve machine learning tasks is crucial to advancing both quantum software and hardware developments. Similarly to the classical setting, the performance of QML models can be significantly affected by the limited access to the underlying data set. Previous studies have focused on proving generalization error bounds for any QML models trained on a limited finite training set. We focus on the optimal QML models obtained by training them on a finite training set and establish a tight prediction error bound in terms of the number of trainable gates and the size of training sets. To achieve this, we derive covering number upper bounds and packing number lower bounds for the data re-uploading QML models and linear QML models, respectively, which may be of independent interest. We support our theoretical findings by numerically simulating the QML strategies for function approximation and quantum phase recognition.