The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. 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 shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new 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.

We give an efficient classical algorithm that recovers the distribution of a non-interacting fermion state over the computational basis. For a system of n non-interacting fermions and m modes, we show that O(m 2 n 4 log(m/δ)/ε 4 ) samples and O(m 4 n 4 log(m/δ)/ε 4 ) time are sufficient to learn the original distribution to total variation distance ε with probability 1 − δ. Our algorithm empirically estimates the one-and two-mode correlations and uses them to reconstruct a succinct description of the entire distribution efficiently.

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