Adaptive estimation of quantum observables

Ariel Shlosberg1,2, Andrew J. Jena3,4, Priyanka Mukhopadhyay3,4, Jan F. Haase3,5,6, Felix Leditzky3,4,7,8, and Luca Dellantonio3,5,9

1JILA, University of Colorado and National Institute of Standards and Technology, Boulder, CO 80309, USA
2Department of Physics, University of Colorado, Boulder, CO 80309, USA
3Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada
4Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada
5Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada
6Institute of Theoretical Physics and IQST, Universität Ulm, D-89069 Ulm, Germany
7Department of Mathematics and IQUIST, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA
8Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
9Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom

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The accurate estimation of quantum observables is a critical task in science. With progress on the hardware, measuring a quantum system will become increasingly demanding, particularly for variational protocols that require extensive sampling. Here, we introduce a measurement scheme that adaptively modifies the estimator based on previously obtained data. Our algorithm, which we call AEQuO, continuously monitors both the estimated average and the associated error of the considered observable, and determines the next measurement step based on this information. We allow both for overlap and non-bitwise commutation relations in the subsets of Pauli operators that are simultaneously probed, thereby maximizing the amount of gathered information. 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 instances. The measurement configuration determined by these subroutines is further post-processed in order to lower the error on the estimator. We test our protocol on chemistry Hamiltonians, for which AEQuO provides error estimates that improve on all state-of-the-art methods based on various grouping techniques or randomized measurements, thus greatly lowering the toll of measurements in current and future quantum applications.

Quantum systems, as opposed to classical ones, are irreversibly destroyed every time they are measured. This has deep implications when one wants to extract information from a quantum system. For instance, when one must estimate the average value of an observable, it is often required to repeat the whole experiment several times. Depending on the measurement strategy employed, the requirements to achieve the same precision vary considerably. In this work, we propose a new approach that considerably lowers the resources on the hardware. Our strategy is adaptive, in the sense that learns and improves the measurement allocation while data is being acquired. Furthermore, it allows for estimating both the average and the error affecting the desired observable at the same time. Compared with other state-of-the-art approaches, we demonstrate consistent and considerable improvement in the accuracy of estimation when our protocol is employed.

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