Classical Shadows with Noise

The classical shadows protocol, recently introduced by Huang, Keung, and Preskill [Nat. Phys. 16, 1050 (2020)], is a hybrid quantum-classical protocol that is used to predict target functions of an unknown quantum state. Unlike full quantum state tomography, the protocol requires only a few quantum measurements to make many predictions with a high success probability, and is therefore more amenable to implementation on near-term quantum hardware. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a generalized shadow norm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define an estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.