  We propose a quantum-classical hybrid variational algorithm, the quantum
orbital minimization method (qOMM), for obtaining the ground state and
low-lying excited states of a Hermitian operator. Given parameterized ansatz
circuits representing eigenstates, qOMM implements quantum circuits to
represent the objective function in the orbital minimization method and adopts
classical optimizer to minimize the objective function with respect to
parameters in ansatz circuits. The objective function has orthogonality
implicitly embedded, which allows qOMM to apply a different ansatz circuit to
each reference state. We carry out numerical simulations that seek to find
excited states of the $\text{H}_{2}$, $\text{LiH}$, and a toy model consisting
of 4 hydrogen atoms arranged in a square lattice in the STO-3G basis and UCCSD
ansatz circuits. Comparing the numerical results with existing excited states
methods, qOMM is less prone to getting stuck in local minima and can achieve
convergence with more shallow ansatz circuits.
