Binary Classification with Classical Instances and Quantum Labels

TL;DR

This paper introduces a new quantum binary classification framework with classical inputs and quantum outputs, demonstrating that its sample complexity matches classical results in key aspects.

Contribution

It proposes a novel quantum classification model with classical-quantum data and establishes tight sample complexity bounds comparable to classical VC-dimension results.

Findings

Sample complexity bounds are tight for pure states.

Quantum model's sample complexity matches classical in dependence on VC-dimension.

Strategies for agnostic and realizable cases are analyzed.

Abstract

In classical statistical learning theory, one of the most well studied problems is that of binary classification. The information-theoretic sample complexity of this task is tightly characterized by the Vapnik-Chervonenkis (VC) dimension. A quantum analog of this task, with training data given as a quantum state has also been intensely studied and is now known to have the same sample complexity as its classical counterpart. We propose a novel quantum version of the classical binary classification task by considering maps with classical input and quantum output and corresponding classical-quantum training data. We discuss learning strategies for the agnostic and for the realizable case and study their performance to obtain sample complexity upper bounds. Moreover, we provide sample complexity lower bounds which show that our upper bounds are essentially tight for pure output states. In particular, we see that the sample complexity is the same as in the classical binary classification task w.r.t. its dependence on accuracy, confidence and the VC-dimension.