On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks
Yizhou Liu, Weijie Su, Tongyang Li
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi></mml:math> 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.