Optimal Parameter Configurations for Sequential Optimization of Variational Quantum Eigensolver

Abstract

Variational Quantum Eigensolver (VQE) is a hybrid algorithm for finding the minimum eigenvalue/vector of a given Hamiltonian by optimizing a parametrized quantum circuit (PQC) using a classical computer. Sequential optimization methods, which are often used in quantum circuit tensor networks, are popular for optimizing the parametrized gates of PQCs. This paper focuses 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 call the parameter configurations. In this study, we first show that the optimization accuracy significantly depends on the choice of parameter configurations due 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.