Classical shadow tomography with locally scrambled quantum dynamics

We generalize the classical shadow tomography scheme to a broad class of finite-depth or finite-time local unitary ensembles, known as locally scrambled quantum dynamics, where ensemble is invariant under local-basis transformations. In this case, reconstruction map for depends only on average entanglement feature snapshots. provide an unbiased estimator state linear combination reduced snapshots in all subsystems, coefficients are solely determined by feature. also bound number experimental measurements required scheme, so-called sample complexity, formulating operator norm formalism. numerically demonstrate our approach circuits and local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve lower complexity compared existing method based Pauli Clifford measurements. Our applicable approximately ensembles with controllable bias that vanishes quickly. Surprisingly, we find single instance time-dependent Hamiltonian evolution sufficient perform approximate it using paradigmatic spin chain modeled after trapped ion Rydberg atom simulators. significantly broadens application near-term devices.