Classical shadow tomography (CST) involves obtaining enough classical descriptions of an unknown state via quantum measurements to predict the outcome of a set of quantum observables. CST has numerous applications, particularly in algorithms that utilize quantum data for tasks such as learning, detection, and optimization. This paper introduces a new CST procedure that exponentially reduces the space complexity and quadratically improves the running time of CST with single-copy measurements. The approach utilizes a quantum-to-classical-to-quantum process to prepare quantum states that represent shadow snapshots, which can then be directly measured by the observables of interest. With that, calculating large matrix traces is avoided, resulting in improvements in running time and space complexity. The paper presents analyses of the proposed methods for CST, with Pauli measurements and Clifford circuits.
Honjani et al. (Tue,) studied this question.