Convergence Analysis of Opto-Electronic Oscillator based Coherent Ising Machines

Ising machines are purported to be better at solving large-scale combinatorial optimisation problems better than conventional von Neumann computers. However, these Ising machines are widely believed to be heuristics, whose promise is observed empirically rather than obtained theoretically. We bridge this gap by considering an opto-electronic oscillator based coherent Ising machine, and providing the first analytical proof that under reasonable assumptions, the OEO-CIM is not a heuristic approach. We find and prove bounds on its performance in terms of the expected difference between the objective value at the final iteration and the optimal one, and on the number of iterations required by it. In the process, we emphasise on some of its limitations such as the inability to handle asymmetric coupling between spins, and the absence of external magnetic field applied on them (both of which are necessary in many optimisation problems), along with some issues in its convergence. We overcome these limitations by proposing suitable adjustments and prove that the improved architecture is guaranteed to converge to the optimum of the relaxed objective function.