  Properties of quantum systems can be estimated using classical shadows, which
implement measurements based on random ensembles of unitaries. Originally
derived for global Clifford unitaries and products of single-qubit Clifford
gates, practical implementations are limited to the latter scheme for moderate
numbers of qubits. Beyond local gates, the accurate implementation of very
short random circuits with two-local gates is still experimentally feasible
and, therefore, interesting for implementing measurements in near-term
applications. In this work, we derive closed-form analytical expressions for
shadow estimation using brickwork circuits with two layers of parallel
two-local Haar-random (or Clifford) unitaries. Besides the construction of the
classical shadow, our results give rise to sample-complexity guarantees for
estimating Pauli observables. We then compare the performance of shadow
estimation with brickwork circuits to the established approach using local
Clifford unitaries and find improved sample complexity in the estimation of
observables supported on sufficiently many qubits.
