Physics-Enhanced Bifurcation Optimisers: All You Need is a Canonical Complex Network

Many physical systems with dynamical evolution that at their steady state give a solution to optimization problems were proposed and realized as promising alternatives to conventional computing. Systems of oscillators such as coherent Ising and XY machines based on lasers, optical parametric oscillators, memristors, polariton and photon condensates are particularly promising due to their scalability, low power consumption and room temperature operation. They achieve a solution via the bifurcation of the fundamental supermode that globally minimizes either the power dissipation of the system or the system Hamiltonian. We show that the canonical Andronov-Hopf networks can capture the bifurcation behaviour of the physical optimizer. Furthermore, a continuous change of variables transforms any physical optimizer into the canonical network so that the success of the physical XY-Ising machine depends primarily on how well the parameters of the networks can be controlled. Our work, therefore, places different physical optimizers in the same mathematical framework that allows for the hybridization of ideas across disparate physical platforms.