Many-body eigenstates from quantum manifold optimization

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 parametrizations 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 parametrizations 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.