Parent Hamiltonian as a benchmark problem for variational quantum eigensolvers

Abstract: 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.