Metastability in Quantum Dynamics for Optimization

Abstract

Metastability separates apparent equilibration from true convergence: a dynamical system can remain near an intermediate state long before reaching its stationary state. In quantum dynamics used for optimization, such a plateau can delay progress toward a target concentrated near the global minimizers of a nonconvex potential. In this paper, we initiate the study of the general mechanisms by which coherent evolution and dissipation shape metastability, using quantum Langevin dynamics (QLD) as a concrete example [CLW+25], where the objective function is encoded in the potential and coherent transfer between wells interacts with tunable dissipation. To quantify this interaction, we solve a two-state model precisely and derive a spectral-separation criterion in terms of tunneling frequency, dissipation strength, and damping imbalance. Extending the analysis to multi-state models, we show that sufficiently weak tunneling between wells can separate slow population transfer from fast relaxation within each well. In addition, we establish conditions under which changing the structure of dissipation shortens the time needed to approach a prescribed target state. Numerical experiments on double-well and multi-well potentials demonstrate these improvements while keeping the target and accuracy fixed and accounting for the time spent on control.