Efficient quantum measurement of Pauli operators in the presence of finite sampling error

Ophelia Crawford1, Barnaby van Straaten1, Daochen Wang1,2, Thomas Parks1, Earl Campbell1,3, and Stephen Brierley1

1Riverlane, Cambridge, UK
2Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, USA
3Department of Physics and Astronomy, University of Sheffield, Sheffield, UK

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Estimating the expectation value of an operator corresponding to an observable is a fundamental task in quantum computation. It is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the operator into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version collects mutually commuting Pauli operators together before measuring all operators within a collection simultaneously. The effectiveness of doing this depends on two factors. Firstly, we must understand the improvement offered by a given arrangement of Paulis in collections. In our work, we propose two natural metrics for quantifying this, operating under the assumption that measurements are distributed optimally among collections so as to minimise the overall finite sampling error. Motivated by the mathematical form of these metrics, we introduce $\large{S}$ORTED $\large{I}$NSERTION, a collecting strategy that exploits the weighting of each Pauli operator in the overall sum. Secondly, to measure all Pauli operators within a collection simultaneously, a circuit is required to rotate them to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any collection of $\boldsymbol{k}$ independent commuting $\boldsymbol{n}$-qubit Pauli operators using at most $\boldsymbol{kn-k(k+1)/2}$ and $\boldsymbol{O(kn/\log k)}$ two-qubit gates respectively. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the operators in question are molecular Hamiltonians. As measured by our metrics, $\large{S}$ORTED $\large{I}$NSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of collections.

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[38] Guillermo García-Pérez, Matteo A. C. Rossi, Boris Sokolov, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Guglielmo Mazzola, Ivano Tavernelli, and Sabrina Maniscalco, "Learning to measure: adaptive informationally complete POVMs for near-term quantum algorithms", arXiv:2104.00569.

[39] Samuel J. Elman, Adrian Chapman, and Steven T. Flammia, "Free fermions behind the disguise", arXiv:2012.07857.

The above citations are from Crossref's cited-by service (last updated successfully 2021-05-06 16:05:48) and SAO/NASA ADS (last updated successfully 2021-05-06 16:05:49). The list may be incomplete as not all publishers provide suitable and complete citation data.

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