Classical shadows over symmetric spaces

TL;DR

This paper extends the theory of classical shadows in quantum computation to symmetric spaces, revealing a unifying framework and potential sample complexity improvements for certain observable estimations.

Contribution

It develops a unifying mathematical theory for classical shadow protocols over symmetric spaces, broadening understanding beyond standard group-based sampling.

Findings

Unifies classical shadow protocols over symmetric spaces.

Identifies potential sample complexity improvements for specific estimations.

Extends the mathematical understanding of classical shadows in quantum computing.

Abstract

Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes.