Classical shadows meet quantum optimal mass transport

TL;DR

This paper introduces a local quantum Wasserstein distance and demonstrates that classical shadows can efficiently estimate quantum states with respect to this distance, especially when prior information is available, impacting quantum generative models.

Contribution

It proposes a local variant of quantum Wasserstein distance and analyzes the efficiency of classical shadows in estimating quantum states under this new metric.

Findings

Classical shadows require O(log n) copies for accurate estimation.

The proposed distance improves understanding of quantum state estimation.

Quantum access is beneficial only with prior information.

Abstract

Classical shadows constitute a protocol to estimate the expectation values of a collection of M observables acting on O(1) qubits of an unknown n-qubit state with a number of measurements that is independent of n and that grows only logarithmically with M. We propose a local variant of the quantum Wasserstein distance of order 1 of [De Palma et al., IEEE Trans. Inf. Theory 67, 6627 (2021)] and prove that the classical shadow obtained measuring O(log n) copies of the state to be learned constitutes an accurate estimate with respect to the proposed distance. We apply the results to quantum generative adversarial networks, showing that quantum access to the state to be learned can be useful only when some prior information on such state is available.