  Extracting information efficiently from quantum systems is a major component
of quantum information processing tasks. Randomized measurements, or classical
shadows, enable predicting many properties of arbitrary quantum states using
few measurements. While random single qubit measurements are experimentally
friendly and suitable for learning low-weight Pauli observables, they perform
poorly for nonlocal observables. Prepending a shallow random quantum circuit
before measurements maintains this experimental friendliness, but also has
favorable sample complexities for observables beyond low-weight Paulis,
including high-weight Paulis and global low-rank properties such as fidelity.
However, in realistic scenarios, quantum noise accumulated with each additional
layer of the shallow circuit biases the results. To address these challenges,
we propose the robust shallow shadows protocol. Our protocol uses Bayesian
inference to learn the experimentally relevant noise model and mitigate it in
postprocessing. This mitigation introduces a bias-variance trade-off:
correcting for noise-induced bias comes at the cost of a larger estimator
variance. Despite this increased variance, as we demonstrate on a
superconducting quantum processor, our protocol correctly recovers state
properties such as expectation values, fidelity, and entanglement entropy,
while maintaining a lower sample complexity compared to the random single qubit
measurement scheme. We also theoretically analyze the effects of noise on
sample complexity and show how the optimal choice of the shallow shadow depth
varies with noise strength. This combined theoretical and experimental analysis
positions the robust shallow shadow protocol as a scalable, robust, and
sample-efficient protocol for characterizing quantum states on current quantum
computing platforms.
