Machine learning of measurement schemes for efficient quantum observable estimation

Estimation of the expectation value of observables is a key subroutine in quantum computing and is also the bottleneck of the performance of many near-term quantum algorithms. Many methods have been proposed to reduce the number of measurements needed for this task by designing measurement schemes that decide the measurements to perform; however, these schemes are usually constructed from hand-crafted heuristics, which limits the measurement efficiency they can achieve. In this paper, we propose a framework for learning measurement schemes directly from the observable, using machine learning techniques including stochastic gradient descent and a two time-scale update rule. As a concrete realization of this framework, we introduce Composite-Locally Biased Classical Shadow (C-LBCS), which learns a mixture of locally-biased classical shadows and their mixing weights end-to-end. We numerically demonstrate C-LBCS on molecular systems up to $\mathrm{CO}_2$ (30 qubits) and show that C-LBCS outperforms the previous state-of-the-art methods despite its simplicity. We believe our approach opens up a reliable and scalable path toward efficient observable estimation on large quantum systems.