Quantum Langevin Dynamics for Optimization

Abstract

We initiate the study of utilizing quantum Langevin dynamics (QLD) to solve optimization problems, particularly those nonconvex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can converge to zero in the low temperature limit with an exponential convergence rate. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with the classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by setting temperature and \(\hbar \) as time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many nonconvex landscapes.