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

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.