Stability of classical shadows under gate-dependent noise

TL;DR

This paper analyzes the stability of classical shadow estimation protocols under gate-dependent noise, showing stability for certain observables and highlighting potential biases for others, with implications for quantum measurement accuracy.

Contribution

It proves stability of Clifford-based shadow estimation under gate-dependent noise for bounded stabilizer norm observables and identifies conditions where noise can cause significant biases.

Findings

Clifford-based shadows are stable under gate-dependent noise for certain observables.

Estimation of 'magic' observables can be highly biased by miscalibration errors.

Robust shadows may introduce large biases under gate-dependent noise.

Abstract

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. In contrast, we demonstrate with concrete examples that estimation of `magic` observables can lead to highly misleading results in the presence of miscalibration errors and a worst case bias scaling 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. Nevertheless, we guarantee the functioning of robust shadows for a more general noise setting than in previous works. 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.