Classical Shadows over Symmetric Spaces

Efficiently learning expectation values of unknown quantum states is a fundamental task in quantum information theory. As the surge in experimentally-accessible quantum systems have forced the development of techniques beyond full state tomography, the theory of classical shadows has emerged as a leading framework within which to perform such learning. In a classical shadows protocol, one measures samples of an unknown state ρ in randomised bases dictated by a protocol-defining ensemble E of unitaries, and uses the resulting data to reconstruct "shadows" of ρ, which are used as proxies to estimate observables. For a given target set of observables, the efficiency of this depends strongly on the selected ensemble of unitaries, with recent heavy interest into the various benefits of different ensembles. Although the protocol is now quite well-understood when E forms a Lie group, there has not been a general understanding of the behaviour of classical shadow protocols when E is not a group. In this work we study the classical shadow protocols induced by sampling uniformly randomly from compact symmetric spaces. Despite the lack of a group structure, we find a unifying theory of such protocols. We find that some of these protocols allow for an improvement in sample-complexity over group-based ensembles for the estimation of observables sampled from certain non-uniform distributions, and may therefore find experimental applications.