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