  Quantum computing offers several new pathways toward finding many-body
eigenstates, with variational approaches being some of the most flexible and
near-term oriented. These require particular parameterizations of the state,
and for solving multiple eigenstates must incorporate orthogonality. In this
work, we use techniques from manifold optimization to arrive at solutions of
the many-body eigenstate problem via direct minimization over the Stiefel and
Grassmannian manifolds, avoiding parameterizations of the states and allowing
for multiple eigenstates to be simultaneously calculated. These Riemannian
manifolds naturally encode orthogonality constraints and have efficient quantum
representations of the states and tangent vectors. We provide example
calculations for quantum many-body molecular systems and discuss different
pathways for solving the multiple eigenstate problem.
