Extra annealing dimensions for improved performance in physical optimizers

Emerging hardware platforms for combinatorial optimization, like the coherent Ising machine (CIM), have established a new class of network dynamics defined by the controlled annealing of an energy landscape from a trivial convex shape to a rugged shape encoding the problem of interest. Their classical and quantum physics have been studied with tools from nonlinear dynamics and disordered systems theory to understand the behavior of both small and large instances. As an optimizer of nonconvex objectives, the CIM faces the challenge of getting trapped in suboptimal local minima. We study two extensions that may help avoid this: adding phase-insensitive gain and allowing asymmetric couplings. The former expands the space of states by providing access to each unit's full complex phase space, while the latter expands the space of connections by making the network a directed graph. Each extension introduces an extra annealing parameter that can be tuned through novel dynamical regimes, recovering conventional CIM operation at one end of its domain. We show how these parameters can be used to redraw bifurcation and phase diagrams and provide paths to global minima in cases where the conventional CIM would get trapped.