Quantifying the Expressive Capacity of Quantum Systems: Fundamental Limits and Eigentasks

TL;DR

This paper develops a mathematical framework to quantify the maximum expressive capacity of quantum systems for machine learning, considering measurement noise, and demonstrates how extracting low-noise eigentasks improves learning performance.

Contribution

It introduces a novel method to evaluate and extract eigentasks, providing a precise bound on quantum expressive capacity and insights into noise reduction via correlations.

Findings

Low-noise eigentasks enhance classification accuracy

Correlations in quantum systems increase learning capacity

Experimental validation on superconducting processors supports theoretical results

Abstract

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.