Randomized Gradient Descents on Riemannian Manifolds: Almost Sure Convergence to Global Minima in and beyond Quantum Optimization

We analyze the convergence properties of gradient descent algorithms on Riemannian manifolds. We study randomization of the tangent space directions of Riemannian gradient flows for minimizing smooth cost functions (of Morse--Bott type) to obtain convergence to local optima. We prove that through randomly projecting Riemannian gradients according to the Haar measure, convergence to local optima can be obtained almost surely despite the existence of saddle points. As an application we consider ground state preparation through quantum optimization over the unitary group. In this setting one can efficiently approximate the Haar-random projections by implementing unitary 2-designs on quantum computers. We prove that the respective algorithm almost surely converges to the global minimum that corresponds to the ground state of a desired Hamiltonian. Finally, we discuss the time required by the algorithm to pass a saddle point in a simple two-dimensional setting.