  Classical shadow tomography provides an efficient method for predicting
functions of an unknown quantum state from a few measurements of the state. It
relies on a unitary channel that efficiently scrambles the quantum information
of the state to the measurement basis. Facing the challenge of realizing deep
unitary circuits on near-term quantum devices, we explore the scenario in which
the unitary channel can be shallow and is generated by a quantum chaotic
Hamiltonian via time evolution. We provide an unbiased estimator of the density
matrix for all ranges of the evolution time. We analyze the sample complexity
of the Hamiltonian-driven shadow tomography. For Pauli observables, we find
that it can be more efficient than the unitary-2-design-based shadow tomography
in a sequence of intermediate time windows that range from an order-1
scrambling time to a time scale of $D^{1/6}$, given the Hilbert space dimension
$D$. In particular, the efficiency of predicting diagonal Pauli observables is
improved by a factor of $D$ without sacrificing the efficiency of predicting
off-diagonal Pauli observables.
