Classical Optimal Control for Energy Minimization Based On Diffeomorphic Modulation under Observable-Response-Preserving Homotopy
In this work, we introduce the so-called “Classical Optimal Control Optimization” (COCO) method for global energy minimization based on the implementation of the diffeomorphic modulation under observable-response-preserving homotopy (DMORPH) gradient algorithm. A probe particle with time-dependent mass $m(t;β)$ and dipole $μ(r,t;β)$ is evolved classically on the potential energy surface $V(r)$ coupled to an electric field $E(t;β)$, as described by the time-dependent density of states represented on a grid, or otherwise as a linear combination of Gaussians generated by the k-means clustering algorithm. Control parameters $β$ defining $m(t;β), μ(r,t;β),$ and $E(t;β)$ are optimized by following the gradients of the energy with respect to $β,$ adapting them to steer the particle toward the global minimum energy configuration. We find that the resulting COCO algorithm is capable of resolving near-degenerate states separated by large energy barriers and successfully locates the global minima of golf potentials on flat and rugged surfaces, previously explored for testing quantum annealing methodologies and the quantum optimal control optimization (QuOCO) method. Preliminary results show successful energy minimization of multidimensional Lennard-Jones clusters. Beyond the analysis of energy minimization in the specific model systems investigated, we anticipate COCO should be valuable for solving minimization problems in general, including optimization of parameters in applications to machine learning and molecular structure determination.