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Classical shadow tomography serves as a potent tool for extracting numerous properties from quantum many-body systems with minimal measurements. Nevertheless, prevailing methods yielding optimal performance for few-body operators necessitate the application of random two-qubit gates, a task that can prove challenging on specific quantum simulators such as ultracold atomic gases. In this work, we introduce an alternative approach founded on the dynamics of many-body localization, a phenomenon extensively demonstrated in optical lattices. Through an exploration of the shadow norm – both analytically, employing a phenomenological model, and numerically, utilizing the TEBD algorithm – we demonstrate that our scheme achieves remarkable efficiency comparable to shallow circuits or measurement-induced criticality, resulting in a significant improvement in the exponential exponent compared to the previous classical shadow protocol. Our findings are corroborated through direct numerical simulations encompassing the entire sampling and reconstruction processes. Consequently, our results present a compelling methodology for analyzing the output states of quantum simulators.
Zhou et al. (Wed,) studied this question.