Stability of classical shadows under gate-dependent noise

Expectation values of observables are routinely estimated using so-called classical shadows$\unicode{x2014}$the outcomes of randomized bases measurements on a repeatedly prepared quantum state. In order to trust the accuracy of shadow estimation in practice, it is crucial to understand the behavior of the estimators under realistic noise. In this work, we prove that any shadow estimation protocol involving Clifford unitaries is stable under gate-dependent noise for observables with bounded stabilizer norm$\unicode{x2014}$originally introduced in the context of simulating Clifford circuits. For these observables, we also show that the protocol's sample complexity is essentially identical to the noiseless case. In contrast, we demonstrate that estimation of `magic' observables can suffer from a bias that scales exponentially in the system size. We further find that so-called robust shadows, aiming at mitigating noise, can introduce a large bias in the presence of gate-dependent noise compared to unmitigated classical shadows. On a technical level, we identify average noise channels that affect shadow estimators and allow for a more fine-grained control of noise-induced biases.