  The variational quantum eigensolver (VQE) and its variants, which is a method
for finding eigenstates and eigenenergies of a given Hamiltonian, are appealing
applications of near-term quantum computers. Although the eigenenergies are
certainly important quantities which determines properties of a given system,
their derivatives with respect to parameters of the system, such as positions
of nuclei if we target a quantum chemistry problem, are also crucial to analyze
the system. Here, we describe methods to evaluate analytical derivatives of the
eigenenergy of a given Hamiltonian, including the excited state energy as well
as the ground state energy, with respect to the system parameters in the
framework of the VQE. We give explicit, low-depth quantum circuits which can
measure essential quantities to evaluate energy derivatives, incorporating with
proof-of-principle numerical simulations. This work extends the theory of the
variational quantum eigensolver, by enabling it to measure more physical
properties of a quantum system than before and to explore chemical reactions.
