On computational capabilities of Ising machines based on nonlinear oscillators,

TL;DR

This paper demonstrates that oscillator networks based on the Kuramoto model can solve Ising problems in polynomial time, highlighting the importance of proper rounding techniques to maintain computational efficiency.

Contribution

It shows that Kuramoto oscillator networks can efficiently approximate Ising ground states with polynomial scaling, linking dynamics to semidefinite programming relaxation.

Findings

Oscillator networks can demonstrate polynomial scaling for Ising problems.

Proper rounding techniques are crucial for maintaining computational capabilities.

Common rounding methods may diminish or invalidate the network's performance.

Abstract

Dynamical Ising machines are actively investigated from the perspective of finding efficient heuristics for NP-hard optimization problems. However, the existing data demonstrate super-polynomial scaling of the running time with the system size, which is incompatible with large NP-hard problems. We show that oscillator networks implementing the Kuramoto model of synchronization are capable of demonstrating polynomial scaling. The dynamics of these networks is related to the semidefinite programming relaxation of the Ising model ground state problem. Consequently, such networks, as we numerically demonstrate, are capable of producing the best possible approximation in polynomial time. To reach such performance, however, the reconstruction of the binary Ising state (rounding) must be specially addressed. We demonstrate that commonly implemented forced collapse to a close-to-Ising state may diminish the computational capabilities up to their complete invalidation. Therefore, consistent treatment of rounding may cardinally improve various operation metrics of already existing and upcoming dynamical Ising machines.