Chasing shadows with Gottesman-Kitaev-Preskill codes

TL;DR

This paper introduces a protocol for shadow tomography of logical subsystems encoded with GKP codes, enabling efficient measurement of quantum states in continuous variable systems without requiring code states.

Contribution

It presents a novel measurement protocol that encodes logical information into classical shadows using twirling, applicable to CV quantum computing measurements like heterodyne and photon parity.

Findings

Protocol enables probabilistic decomposition into Gaussian states.

Develops sampling strategies for Wigner sampling protocol.

Allows estimation of bounded observables via randomized GKP codes.

Abstract

We consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the protocol yields a probabilistic decomposition of any input state into Gaussian states that simulate the encoded logical information of the input relative to a fixed GKP code where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.