Statistical Complexity of Quantum Learning

TL;DR

This paper reviews the information-theoretic complexity of quantum learning, focusing on data, copy, and model complexities, and highlights fundamental differences from classical learning due to quantum measurement constraints.

Contribution

It provides a comprehensive analysis of quantum learning complexity using information-theoretic techniques, including background and distinctions from classical learning.

Findings

Quantum measurements limit data extraction due to irreversibility.

Quantum learning involves unique complexities like copy complexity.

The paper offers extensive background and literature pointers.

Abstract

Recent years have seen significant activity on the problem of using data for the purpose of learning properties of quantum systems or of processing classical or quantum data via quantum computing. As in classical learning, quantum learning problems involve settings in which the mechanism generating the data is unknown, and the main goal of a learning algorithm is to ensure satisfactory accuracy levels when only given access to data and, possibly, side information such as expert knowledge. This article reviews the complexity of quantum learning using information-theoretic techniques by focusing on data complexity, copy complexity, and model complexity. Copy complexity arises from the destructive nature of quantum measurements, which irreversibly alter the state to be processed, limiting the information that can be extracted about quantum data. For example, in a quantum system, unlike in classical machine learning, it is generally not possible to evaluate the training loss simultaneously on multiple hypotheses using the same quantum data. To make the paper self-contained and approachable by different research communities, we provide extensive background material on classical results from statistical learning theory, as well as on the distinguishability of quantum states. Throughout, we highlight the differences between quantum and classical learning by addressing both supervised and unsupervised learning, and we provide extensive pointers to the literature.