



02114nas a2200181   4500008004100000245005300041210005300094260001500147490000600162520160200168100002201770700002601792700001801818700001801836700001901854700002201873856003701895       2021                        eng d00aEfficient quantum measurement of Pauli operators0 aEfficient quantum measurement of Pauli operators  c01/19/20210 v53 a<p>Estimating the expectation value of an observable is a fundamental task in quantum computation. Unfortunately, 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 observable into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version first groups mutually commuting Pauli operators together and then measures all operators within each group simultaneously. The effectiveness of this depends on two factors. First, to enable simultaneous measurement, circuits are required to rotate each group to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any group of k commuting n-qubit Pauli operators using at most kn&minus;k(k+1)/2 and O(kn/logk) two-qubit gates respectively. Second, metrics that justifiably measure the effectiveness of a grouping are required. In our work, we propose two natural metrics that operate under the assumption that measurements are distributed optimally among groups. Motivated by our new metrics, we introduce SORTED INSERTION, a grouping strategy that is explicitly aware of the weighting of each Pauli operator in the observable. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the observables in question are molecular Hamiltonians. As measured by our metrics, SORTED INSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of groups.</p>1 aCrawford, Ophelia1 avan Straaten, Barnaby1 aWang, Daochen1 aParks, Thomas1 aCampbell, Earl1 aBrierley, Stephen  uhttps://arxiv.org/abs/1908.06942