A spin Hall Ising machine.*

Ising Machines (IMs) are physical systems designed to find solutions to combinatorial optimization (CO) problems mapped onto the IM via the coupling strengths of its binary spins. Using the intrinsic dynamics and different annealing schemes, the IM relaxes over time to its lowest energy state, which is the solution to the CO problem. Here we present the world's most miniaturized, and the first spin Hall nano-oscillator (SHNO) based Ising machine. One of the most intriguing properties of SHNOs is their ability to synchronize to each other and to an external source. We demonstrate robust phase binarization due to injection locking at twice the natural frequency of SHNOs and analyze the binarization behavior as a function of injection locking power and frequency.  We then show binarization in 1x2 and 2x2 SHNO interconnected arrays. The phase binarization manifests itself as distinct microwave output power levels, which are readily distinguished using electrical means. In addition, we use phase-resolved Brillouin Light Scattering (phase-BLS) microscopy to directly observe the individual phases of the precessing magnetization in each nano-constriction. The different states can be accessed using either different injected power levels or a detuned frequency of the injected signal. We then propose pathways to control the coupling between SHNOs in order to gain flexibility in terms of mapped problems, and realize truly miniaturized, ultra-fast, and large-scale oscillator-based Ising Machines.