  Variational quantum eigensolver (VQE), which attracts attention as a
promising application of noisy intermediate-scale quantum devices, finds a
ground state of a given Hamiltonian by variationally optimizing the parameters
of quantum circuits called ansatz. Since the difficulty of the optimization
depends on the complexity of the problem Hamiltonian and the structure of the
ansatz, it has been difficult to analyze the performance of optimizers for the
VQE systematically. To resolve this problem, we propose a technique to
construct a benchmark problem whose ground state is guaranteed to be achievable
with a given ansatz by using the idea of parent Hamiltonian of low-depth
parameterized quantum circuits. We compare the convergence of several
optimizers by varying the distance of the initial parameters from the solution
and find that the converged energies showed a threshold-like behavior depending
on the distance. This work provides a systematic way to analyze optimizers for
VQE and contribute to the design of ansatz and its initial parameters.
