Classical shadows with arbitrary group representations

TL;DR

This paper develops a unified theoretical framework for classical shadows in quantum state prediction, extending to general group representations and introducing new protocols with minimized classical post-processing.

Contribution

It generalizes previous classical shadow methods to arbitrary group representations, identifying centralizing bases for efficient measurement channel inversion and sample complexity bounds.

Findings

Unified theory for CS protocols based on general group representations

Introduction of centralizing bases for measurement channel inversion

Characterization of new shadow protocols using various group representations

Abstract

Classical shadows (CS) has recently emerged as an important framework to efficiently predict properties of an unknown quantum state. A common strategy in CS protocols is to parametrize the basis in which one measures the state by a random group action; many examples of this have been proposed and studied on a case-by-case basis. In this work, we present a unified theory that allows us to simultaneously understand CS protocols based on sampling from general group representations, extending previous approaches that worked in simplified (multiplicity-free) settings. We identify a class of measurement bases which we call "centralizing bases" that allows us to analytically characterize and invert the measurement channel, minimizing classical post-processing costs. We complement this analysis by deriving general bounds on the sample-complexity necessary to obtain estimates of a given precision. Beyond its unification of previous CS protocols, our method allows us to readily generate new protocols based on other groups, or different representations of previously considered ones. For example, we characterize novel shadow protocols based on sampling from the spin and tensor representations of $\textsf{SU}(2)$, symmetric and orthogonal groups, and the exceptional Lie group $G_2$.