Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. We investigate the task of partitioning a random subset of Pauli operators into simultaneously-measurable parts. Using heuristics from coloring random graphs, we give an upper bound for the expected number of parts in our partition. We go on to conjecture that allowing arbitrary Clifford operators before measurement, rather than single-qubit operations, leads to a decrease in the number of parts which is linear with respect to the lengths of the operators. We give evidence to confirm this conjecture and comment on the importance of this result for a specific near-term application: speeding up the measurement process of the variational quantum eigensolver.

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The accurate estimation of quantum observables is a fundamental task in science. Here, we introduce a measurement scheme that adaptively modifies the estimator based on previously obtained data. Our algorithm, which we call AEQuO, allows for overlap in the subsets of Pauli operators that are measured simultaneously, thereby maximizing the amount of information gathered in the measurements. The adaptive estimator 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 as well as many-body Hamiltonians with different interaction ranges. In all cases, our algorithm provides error estimates that improve on state-of-the-art methods based on various grouping techniques or randomized measurements.

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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.