Generalization Error in Quantum Machine Learning in the Presence of Sampling Noise

TL;DR

This paper analyzes the impact of sampling noise on quantum machine learning, providing a statistical mechanics framework to quantify generalization errors and demonstrating Eigentask Learning's optimality in noisy, finite data scenarios.

Contribution

It introduces a statistical mechanics approach to quantify generalization errors in quantum machine learning with finite data and sampling noise, validating Eigentask Learning's effectiveness.

Findings

Quantitative characterization of generalization errors with finite data and noise.

Analytical and numerical validation of Eigentask Learning's optimality.

Demonstration that Eigentask Learning minimizes generalization errors.

Abstract

Tackling output sampling noise due to finite shots of quantum measurement is an unavoidable challenge when extracting information in machine learning with physical systems. A technique called Eigentask Learning was developed recently as a framework for learning with infinite input training data in the presence of output sampling noise. In the work of Eigentask Learning, numerical evidence was presented that extracting low-noise contributions of features can practically improve performance for machine learning tasks, displaying robustness to overfitting and increasing generalization accuracy. However, it remains unsolved to quantitatively characterize generalization errors in situations where the training dataset is finite, while output sampling noise still exists. In this study, we use methodologies from statistical mechanics to calculate the training and generalization errors of a generic quantum machine learning system when the input training dataset and output measurement sampling shots are both finite. Our analytical findings, supported by numerical validation, offer solid justification that Eigentask Learning provides optimal learning in the sense of minimizing generalization errors.