  The expressive capacity of quantum systems for machine learning is limited by
quantum sampling noise incurred during measurement. Although it is generally
believed that noise limits the resolvable capacity of quantum systems, the
precise impact of noise on learning is not yet fully understood. We present a
mathematical framework for evaluating the available expressive capacity of
general quantum systems from a finite number of measurements, and provide a
methodology for extracting the extrema of this capacity, its eigentasks.
Eigentasks are a native set of functions that a given quantum system can
approximate with minimal error. We show that extracting low-noise eigentasks
leads to improved performance for machine learning tasks such as
classification, displaying robustness to overfitting. We obtain a tight bound
on the expressive capacity, and present analyses suggesting that correlations
in the measured quantum system enhance learning capacity by reducing noise in
eigentasks. These results are supported by experiments on superconducting
quantum processors. Our findings have broad implications for quantum machine
learning and sensing applications.
