Quantum algorithm for finding the negative curvature direction in non-convex optimization

TL;DR

This paper introduces a quantum algorithm that efficiently finds negative curvature directions in non-convex optimization, significantly outperforming classical methods in speed and scalability.

Contribution

The paper presents a novel quantum algorithm for negative curvature detection with exponentially improved query complexity over classical approaches.

Findings

Quantum algorithm achieves query complexity O(polylog(d)/ε).

Exponential speedup over classical methods in finding negative curvature.

Efficient quantum read-out algorithm surpasses classical counterparts.

Abstract

We present an efficient quantum algorithm aiming to find the negative curvature direction for escaping the saddle point, which is the critical subroutine for many second-order non-convex optimization algorithms. We prove that our algorithm could produce the target state corresponding to the negative curvature direction with query complexity O(polylog(d) /{\epsilon}), where d is the dimension of the optimization function. The quantum negative curvature finding algorithm is exponentially faster than any known classical method which takes time at least O(d /\sqrt{\epsilon}). Moreover, we propose an efficient quantum algorithm to achieve the classical read-out of the target state. Our classical read-out algorithm runs exponentially faster on the degree of d than existing counterparts.