Stability of Classical Shadows under Gate-Dependent Noise

Expectation values of observables are routinely estimated using so-called classical shadows—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 Letter, we prove that any shadow estimation protocol involving Clifford unitaries is stable under gate-dependent noise for observables with bounded stabilizer norm—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-rained control of noise-induced biases.

DDC Class

600: Technology

Funding(s)

DFG Grant No. 441423094, BMBF Grant No. 13N15522

Funding Organisations

Deutsche Forschungsgemeinschaft (DFG)

More Funding Information

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Emmy Noether program (Grant No. 441423094), the German Federal Ministry of Education and Research (BMBF) within the funding program “Quantum technologies—from basic research to market” in the joint project MIQRO (Grant No. 13N15522), and by the Fujitsu Germany GmbH as part of the endowed professorship “Quantum Inspired and Quantum Optimization”.

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Stability of Classical Shadows under Gate-Dependent Noise