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

arXiv: https://arxiv.org/abs/2405.12039 · HTML: https://mlsanity.com/paper/2405.12039/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2024.

## Built on by (most cited first, 1 of 1)

- [Riemannian Optimization for Multi-Player Quantum Games on Product Unitary Manifolds](https://mlsanity.com/paper/2609.07397/) (2026)

## Abstract (arXiv)

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 → operatornametr(AUρ U^*) almost surely converges to its global minimum. The minimum corresponds to the smallest eigenvalue (ground state) of the selfadjoint operator A (Hamiltonian) if ρ 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.