The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision ε, QPE requires O(1) repetitions of circuits with depth O(1/ε), whereas each expectation estimation subroutine within VQE requires O(1/ε2) samples from circuits with depth O(1). We propose a generalised VQE algorithm that interpolates between these two regimes via a free parameter α∈[0,1] which can exploit quantum coherence over a circuit depth of O(1/εα) to reduce the number of samples to O(1/ε2(1−α)). Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.

1 aWang, Daochen1 aHiggott, Oscar1 aBrierley, Stephen uhttps://arxiv.org/abs/1802.0017102096nas a2200169 4500008004100000245005300041210005300094260001500147520160200162100002201764700002601786700001801812700001801830700001901848700002201867856003701889 2019 eng d00aEfficient quantum measurement of Pauli operators0 aEfficient quantum measurement of Pauli operators c08/19/20193 aEstimating 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−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.

1 aCrawford, Ophelia1 avan Straaten, Barnaby1 aWang, Daochen1 aParks, Thomas1 aCampbell, Earl1 aBrierley, Stephen uhttps://arxiv.org/abs/1908.0694201313nas a2200145 4500008004100000245005400041210005400095260001500149490000600164520090100170100001901071700001801090700002201108856003701130 2019 eng d00aVariational Quantum Computation of Excited States0 aVariational Quantum Computation of Excited States c06/28/20190 v33 aThe calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum compute

1 aHiggott, Oscar1 aWang, Daochen1 aBrierley, Stephen uhttps://arxiv.org/abs/1805.08138