  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.
