  The calculation of off-diagonal matrix elements has various applications in
fields such as nuclear physics and quantum chemistry. In this paper, we present
a noisy intermediate scale quantum algorithm for estimating the diagonal and
off-diagonal matrix elements of a generic observable in the energy eigenbasis
of a given Hamiltonian. Several numerical simulations indicate that this
approach can find many of the matrix elements even when the trial functions are
randomly initialized across a wide range of parameter values without, at the
same time, the need to prepare the energy eigenstates.
