Optimal parameter configurations for sequential optimization of the variational quantum eigensolver

Abstract

The variational quantum eigensolver (VQE) is a hybrid algorithm to find the minimum eigenvalue/vector of the given Hamiltonian by optimizing a parameterized quantum circuit (PQC) using a classical computer. Sequential optimization methods, which are often used in quantum circuit tensor networks, are popular for optimizing the parameterized gates of PQCs. In this paper, we focus on the case where the components to be optimized are single-qubit gates, in which the analytic optimization of a single-qubit gate is sequentially performed. The analytical solution is given by diagonalization of a matrix whose elements are computed from the expectation values of observables specified by a set of predetermined parameters, which we refer to as the parameter configurations. In this paper, we first show that the optimization accuracy significantly depends on the choice of parameter configurations owing to the statistical errors in the expectation values. We then identify a metric that quantifies the optimization accuracy of a parameter configuration for all possible statistical errors, named configuration overhead/cost or C-cost. We theoretically provide the lower bound of C-cost and show that, for the minimum size of parameter configurations, the lower bound is achieved if and only if the parameter configuration satisfies the so-called equiangular line condition. Finally, we provide numerical experiments demonstrating that the optimal parameter configuration exhibits the best result in several VQE problems. We hope that this general statistical methodology will enhance the efficacy of sequential optimization of PQCs for solving practical problems with near-term quantum devices.