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Combinatorial optimization problems are known for being particularly hard to solve on traditional von Neumann architectures. This has led to the development of Ising Machines (IMs) based on quantum annealers and optical and electronic oscillators, demonstrating speed-ups compared to central processing unit (CPU) and graphics processing unit (GPU) algorithms. Spin torque nano-oscillators (STNOs) have shown GHz operating frequency, nanoscale size, and nanosecond turn-on time, which would allow their use in ultrafast oscillator-based IMs. Here, we show using numerical simulations based on STNO auto-oscillator theory that STNOs exhibit fundamental characteristics needed to realize IMs, including in-phase/out-of-phase synchronization and second harmonic injection locking phase binarization. Furthermore, we demonstrate numerically that large STNO network IMs can solve Max-Cut problems on nanosecond timescales., QC 20210419

In magnetic trilayer systems, spin pumping is generally addressed as a reciprocal mechanism characterized by one unique spin-mixing conductance common to both interfaces. However, this assumption is questionable in cases where different types of interfaces are present. Here, we present a general theory for analyzing spin pumping in cases with more than one unique interface and where the magnetic coupling is allowed to be noncollinear. The theory is applied to analyze layer-resolved ferromagnetic resonance experiments on the trilayer system Ni80Fe20/Ru/Fe49Co49V2 where the Ru spacer thickness is varied to tune the indirect exchange coupling. It is demonstrated that the equation of motion of macrospins driven by spin pumping need to be modified in case of noncollinear coupling. Our analysis also shows that the spin pumping in trilayer systems with dissimilar magnetic layers, in general, is nonreciprocal.