  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.
