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

Tackling sampling noise is an unavoidable challenge when extracting information in machine learning with physical systems. Eigentask Learning was developed in recent work as a framework for learning in the presence of sampling noise [1]. In that work, numerical evidence was presented that extracting low-noise eigentasks can lead to improved performance for machine learning tasks displaying robustness to overfitting, increasing generalization accuracy. The issue of characterizing generalization errors in situations where the training dataset is finite remains unresolved. In this study, we employ methodologies from statistical mechanics to calculate the learning curve of a generic quantum machine learning system, when the training set contains a large but finite number of samples. Our analytical findings, supported by numerical validation, offer solid justification that Eigentask Learning provides optimal learning, in the sense of minimizing generalization errors.