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.

翻译：经典算法通常难以有效解决局部极小值被高势垒分隔的非凸优化问题。本文通过利用量子隧穿的全局效应，探索非凸优化可能的量子加速途径。具体而言，我们提出了一种称为量子隧穿行走（QTW）的量子算法，并将其应用于局部极小值近似为全局极小值的非凸问题。研究表明，当不同局部极小值之间的势垒高而窄且极小值平坦时，QTW相较于经典随机梯度下降（SGD）实现了量子加速。基于这一发现，我们构建了一个特定的双势阱景观：已知一个势阱时经典算法无法有效击中目标势阱，而给定靠近已知势阱的适当初始状态时QTW能够实现这一目标。最后，我们通过数值实验验证了上述结论。