  The variational quantum eigensolver (VQE) is an algorithm to find
eigenenergies and eigenstates of systems in quantum chemistry and quantum
many-body physics. The VQE is one of the most promising applications of
near-term quantum devices to investigate such systems. Here we propose an
extension of the VQE to calculate the nonadiabatic couplings of molecules in
quantum chemical systems and Berry's phase in quantum many-body systems. Both
quantities play an important role to understand the properties of a system
beyond the naive adiabatic picture, e.g., nonadiabatic dynamics and topological
phase of matter. We provide quantum circuits and classical post-processings to
calculate the nonadiabatic couplings and Berry's phase. Specifically, we show
that the evaluation of the nonadiabatic couplings reduces to that of
expectation values of observables while that of Berry's phase also requires one
additional Hadamard test. Furthermore, we simulate the photodissociation
dynamics of a lithium fluoride molecule using the nonadiabatic couplings
evaluated on a real quantum device. Our proposal widens the applicability of
the VQE and the possibility of near-term quantum devices to study molecules and
quantum many-body systems.
