Adaptive estimation of quantum observables

The accurate estimation of quantum observables is a critical task in science. With progress on the hardware, measur-ing a quantum system will become increas-ingly demanding, particularly for vari-ational protocols that require extensive sampling. Here, we introduce a mea-surement scheme that adaptively modi-fies the estimator based on previously ob-tained data. Our algorithm, which we call AEQuO, continuously monitors both the estimated average and the associated er-ror of the considered observable, and de-termines the next measurement step based on this information. We allow both for overlap and non-bitwise commutation re-lations in the subsets of Pauli operators that are simultaneously probed, thereby maximizing the amount of gathered infor-mation. AEQuO comes in two variants: a greedy bucket-filling algorithm with good performance for small problem instances, and a machine learning-based algorithm with more favorable scaling for larger in-stances. The measurement configuration determined by these subroutines is further post-processed in order to lower the er-ror on the estimator. We test our proto-col on chemistry Hamiltonians, for which AEQuO provides error estimates that im-prove on all state-of-the-art methods based on various grouping techniques or random-ized measurements, thus greatly lowering the toll of measurements in current and future quantum applications.