  Calculating ground and excited states is an exciting prospect for near-term
quantum computing applications, and accurate and efficient algorithms are
needed to assess viable directions. We develop an excited state approach based
on the contracted quantum eigensolver (ES-CQE), which iteratively attempts to
find a solution to a contraction of the Schr{\"o}dinger equation projected onto
a subspace, and does not require a priori information on the system. We focus
on the anti-Hermitian portion of the equation, leading to a two-body unitary
ansatz. We investigate the role of symmetries, initial states, constraints, and
overall performance within the context of the model rectangular ${\rm H}_4$
system. We show the ES-CQE achieves near-exact accuracy across the majority of
states, covering regions of strong and weak electron correlation, while also
elucidating challenging instances for two-body unitary ansatz.
