Quantum algorithms for escaping from saddle points

TL;DR

This paper introduces a quantum algorithm for efficiently escaping saddle points in optimization problems, outperforming classical methods in terms of query complexity by leveraging quantum wave simulations.

Contribution

The paper presents the first quantum algorithm for escaping saddle points with provable guarantees, replacing classical perturbations with quantum wave equations.

Findings

Quantum algorithm uses fewer queries than classical counterparts.

Replaces classical perturbations with quantum wave simulations.

Numerical experiments support theoretical improvements.

Abstract

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function $f\colon\mathbb{R}^{n}\to\mathbb{R}$, our quantum algorithm outputs an $\epsilon$-approximate second-order stationary point using $\tilde{O}(\log^{2} (n)/\epsilon^{1.75})$ queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with $\tilde{O}(\log^{6} (n)/\epsilon^{1.75})$ queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of $\log n$ and matches its complexity in terms of $1/\epsilon$. Technically, our main contribution is the idea of replacing the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the improvement in the quantum query complexity with $\log n$ factors for escaping from saddle points. We also show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our theoretical findings.