Steady state distributions of network of degenerate optical parametric oscillators in solving combinatorial optimization problems

TL;DR

This paper models a network of degenerate optical parametric oscillators as a coherent Ising machine, deriving steady state distributions that identify optimal solutions for certain Ising problems, including phase transitions.

Contribution

It extends the coherent Ising machine model to multiple DOPOs and analytically derives steady state distributions for solving Ising problems.

Findings

Steady state distributions correspond to optimal solutions in simple Ising problems.

The model detects phase transitions in random-field Ising models.

Analytical results match expected behavior in specific problem settings.

Abstract

We investigate network of degenerate optical parametric oscillators (DOPOs) as a model of the coherent Ising machine, an architecture for solving Ising problems. The network represents the interaction in the Ising model, which is a generalization of a previously proposed one for the two-DOPO case. Dynamics of the DOPOs is described by the Fokker-Planck equation in the positive $P$ representation. We obtain approximate steady state distributions for arbitrary Ising problems under some ansatz. Using the method of statistical mechanics, we analytically demonstrate that the most probable states in a particular range of the parameters correspond to the true optimal states for two rather simple problems, i.e., fully-connected ferromagnetic coupling without/with binary random fields. In particular, for the random-field problem, the distribution correctly detects the phase transition that occurs in the target Ising model with varying the magnitude of the fields.