Phase analysis of Ising machines and their implications on optimization

Ising machines, which are dynamical systems designed to operate in a parallel and iterative manner, have emerged as a paradigm for solving combinatorial optimization problems. Despite computational advantages, the quality of solutions depends heavily on the form of dynamics and tuning of parameters, which are, in general, set heuristically due to the lack of systematic insights. Here, we focus on optimal Ising machine design by analyzing phase diagrams of spin distributions in the Sherrington-Kirkpatrick model. We find that the ground state can be achieved in the phase where the spin distribution becomes binary, and optimal solutions are produced where the binary phase and gapless phase coexist. Our analysis shows that such coexistence phase region can be expanded by carefully placing a digitization operation, giving rise to a family of superior Ising machines, as illustrated by the proposed algorithm digCIM.