On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks

TL;DR

This paper introduces a quantum tunneling walk algorithm that offers potential quantum speedups for nonconvex optimization problems, especially when local minima are separated by high, thin barriers, demonstrated through theoretical analysis and numerical experiments.

Contribution

The paper proposes the quantum tunneling walk (QTW) algorithm, a novel quantum approach that leverages tunneling effects to outperform classical methods in specific nonconvex landscapes.

Findings

QTW achieves quantum speedup over classical SGD in certain nonconvex problems.

QTW can efficiently locate targets in double-well landscapes where classical algorithms struggle.

Numerical experiments support the theoretical advantages of QTW in relevant scenarios.

Abstract

Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.