Generalization of the output of variational quantum eigensolver by parameter interpolation with low-depth ansatz

The variational quantum eigensolver (VQE) is an attracting possible application of near-term quantum computers. Originally, the aim of the VQE is to find a ground state for a given specific Hamiltonian. It is achieved by minimizing the expectation value of the Hamiltonian with respect to an ansatz state by tuning parameters bmθ on a quantum circuit which constructs the ansatz. Here we consider an extended problem of the VQE, namely, our objective in this work is to "generalize" the optimized output of the VQE just like machine learning. We aim to find ground states for a given set of Hamiltonians \{H\(bm{x}\}\), where bm{x} is a parameter which specifies the quantum system under consideration, such as geometries of atoms of a molecule. Our approach is to train the circuit on the small number of bm{x}'s. Specifically, we employ the interpolation of the optimal circuit parameter determined at different bm{x}'s, assuming that the circuit parameter bmθ has simple dependency on a hidden parameter bm{x} as bmθ\(bm{x}\). We show by numerical simulations that, using an ansatz which we call the Hamiltonian-alternating ansatz, the optimal circuit parameters can be interpolated to give near-optimal ground states in between the trained bm{x}'s. The proposed method can greatly reduce, on a rough estimation by a few orders of magnitude, the time required to obtain ground states for different Hamiltonians by the VQE. Once generalized, the ansatz circuit can predict the ground state without optimizing the circuit parameter bmθ in a certain range of bm{x}.