A global Lyapunov function for the coherent Ising machine

Two classical Ising machine schemes, the Oscillator Ising Machine (OIM) and the Bistable Latch Ising Machine (BLIM), have been shown to feature global Lyapunov functions, i.e., continuous "energy-like" functions whose local minima are naturally found by the physics of these schemes. We show that the Coherent Ising Machine (CIM), an optical scheme that predated OIM and BLIM, also has a global Lyapunov function that approximates the Ising Hamiltonian at stable equilibrium points. Our result sharpens understanding of CIM operation, revealing that its mechanism for breaking out of local minima is a purely probabilistic classical one, similar to Gibbs sampling.