  A balanced description of both static and dynamic correlations in electronic
systems with nearly degenerate low-lying states presents a challenge for
multi-configurational methods on classical computers. We present here a quantum
algorithm utilizing the action of correlating cluster operators to provide
high-quality wavefunction ans\"atze employing a non-orthogonal multireference
basis that captures a significant portion of the exact wavefunction in a highly
compact manner, and that allows computation of the resulting energies and
wavefunctions at polynomial cost with a quantum computer. This enables a
significant improvement over the corresponding classical non-orthogonal solver,
which incurs an exponential cost when evaluating off-diagonal matrix elements
between the ansatz states, and is therefore intractable. We implement the
non-orthogonal quantum eigensolver (NOQE) here with an efficient ansatz
parameterization inspired by classical quantum chemistry methods that succeed
in capturing significant amounts of electronic correlation accurately. By
taking advantage of classical methods for chemistry, NOQE provides a flexible,
compact, and rigorous description of both static and dynamic electronic
correlation, making it an attractive method for the calculation of electronic
states of a wide range of molecular systems.
