Stability of classical shadows 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.

26 pages, 1 figure; v2: Improved presentation of results in the main text; v3: More concise main text and new explanatory example using two additional figures

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• Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows
• Probing quantum complexity via universal saturation of stabilizer entropies
• Operational interpretation of the Stabilizer Entropy
• Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks
• Quantum-Classical Auxiliary Field Quantum Monte Carlo with Matchgate Shadows on Trapped Ion Quantum Computers
• Shot-noise reduction for lattice Hamiltonians
• Classical Shadows with Improved Median-of-Means Estimation