Phase-Binarized Spintronic Oscillators for Combinatorial Optimization, and Comparison with Alternative Classical and Quantum Methods

Solving combinatorial optimization problems efficiently through emerging hardware by converting the problem to its equivalent Ising model and obtaining its ground state is known as Ising computing. Phase-binarized oscillators (PBO), modeled through the Kuramoto model, have been proposed for Ising computing, and various device technologies have been used to experimentally implement such PBOs. In this paper, we show that an array of four dipole-coupled uniform-mode spin Hall nano oscillators (SHNOs) can be used to implement such PBOs and solve the NP-Hard combinatorial problem MaxCut on 4-node complete weighted graphs. We model the spintronic oscillators through two techniques: an approximate model for coupled magnetization dynamics of spin oscillators, and Landau Lifshitz Gilbert Slonckzweski (LLGS) equation-based more accurate magnetization dynamics modeling of such oscillators. Next, we compare the performance of these room-temperature-operating spin oscillators, as well as generalized PBOs, with two other alternative methods that solve the same MaxCut problem: a classical approximation algorithm, known as Goemans-Williamson's (GW) algorithm, and a Noisy Intermediate Scale Quantum (NISQ) algorithm, known as Quantum Approximation Optimization Algorithm (QAOA). For four types of graphs, with graph size up to twenty nodes, we show that approximation ratio (AR) and success probability (SP) obtained for generalized PBOs (Kuramoto model), as well as spin oscillators, are comparable to that for GW and much higher than that of QAOA for almost all graph instances. Moreover, unlike GW, the time to solution (TTS) for generalized PBOs and spin oscillators does not grow with graph size for the instances we have explored. This can be a major advantage for PBOs in general and spin oscillators specifically for solving these types of problems, along with the accuracy of solutions they deliver.