  We propose a hybrid quantum-classical algorithm for approximating the ground
state and ground state energy of a Hamiltonian. Once the Ansatz has been
decided, the quantum part of the algorithm involves the calculation of two
overlap matrices. The output from the quantum part of the algorithm is utilized
as input for the classical computer. The classical part of the algorithm is a
quadratically constrained quadratic program with a single quadratic equality
constraint. Unlike the variational quantum eigensolver algorithm, our algorithm
does not have any classical-quantum feedback loop. Using convex relaxation
techniques, we provide an efficiently computable lower bound to the classical
optimization program. Furthermore, using results from Bar-On et al. (Journal of
Optimization Theory and Applications, 82(2):379--386, 1994), we provide a
sufficient condition for a local minimum to be a global minimum. A solver can
use such a condition as a stopping criterion.
