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.

%B Phys. Rev. Lett. %V 122 %8 3/25/2019 %G eng %U https://arxiv.org/abs/1802.00171 %N 140504 %R https://doi.org/10.1103/PhysRevLett.122.140504 %0 Journal Article %J Quantum %D 2019 %T Variational Quantum Computation of Excited States %A Oscar Higgott %A Daochen Wang %A Stephen Brierley %XThe 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

%B Quantum %V 3 %8 06/28/2019 %G eng %U https://arxiv.org/abs/1805.08138 %N 156 %R https://doi.org/10.22331/q-2019-07-01-156