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

TL;DR

This paper proves that randomized gradient descent methods on Riemannian manifolds almost surely converge to global minima, with applications to quantum optimization and ground-state preparation, even in the presence of saddle points.

Contribution

It introduces a randomized gradient descent algorithm on Riemannian manifolds that guarantees convergence to global minima, including discrete implementations for quantum optimization.

Findings

Almost sure convergence to global minima despite saddle points

Effective use of discrete unitary 2-designs in quantum optimization

Analysis of saddle point traversal time

Abstract

We analyze convergence of gradient-descent methods on Riemannian manifolds. In particular, we study randomization of Riemannian gradient algorithms for minimizing smooth cost functions (of Morse-Bott type). We prove that randomized gradient descent methods, where the Riemannian gradient is replaced by a random projection of it, converge to a single local optimum almost surely despite the existence of saddle points. We consider both uniformly distributed and discrete random projections. We also discuss the time required to pass a saddle point. As a major application, we consider ground-state preparation through quantum optimization over the unitary group. In mathematical terms our randomized algorithm applied to the trace function $U \to \operatorname{tr}(AU\rho U^*)$ almost surely converges to its global minimum. The minimum corresponds to the smallest eigenvalue (ground state) of the selfadjoint operator $A$ (Hamiltonian) if $\rho$ is a rank-one projector (pure state). In this setting, one can efficiently replace the uniform random projections by implementing so-called discrete unitary 2-designs.