  We propose a state-averaged orbital optimization scheme for improving the
accuracy of excited states of the electronic structure Hamiltonian for use on
near-term quantum computers. Instead of parameterizing the orbital rotation
operator in the conventional fashion as an exponential of an anti-hermitian
matrix, we parameterize the orbital rotation as a general partial unitary
matrix. Whereas conventional orbital optimization methods minimize the
state-averaged energy using successive Newton steps of the second-order Taylor
expansion of the energy, the method presented here optimizes the state-averaged
energy using an orthogonally-constrained gradient projection method which does
not require any expansion approximations. Through extensive benchmarking of the
method on various small molecular systems, we find that the method is capable
of producing more accurate results than fixed basis FCI while simultaneously
using fewer qubits. In particular, we show that for $\mathrm{H_2}$, the method
is capable of matching the accuracy of FCI in the cc-pVTZ basis (56 qubits)
while only using 14 qubits.
