  Computationally hard problems, including combinatorial optimization, can be
mapped into the problem of finding the ground-state of an Ising Hamiltonian.
Building physical systems with collective computational ability and distributed
parallel processing capability can accelerate the ground-state search. Here, we
present a continuous-time dynamical system (CTDS) approach where the
ground-state solution appears as stable points or attractor states of the CTDS.
We harness the emergent dynamics of a network of phase-transition
nano-oscillators (PTNO) to build an Ising Hamiltonian solver. The hardware
fabric comprises of electrically coupled injection-locked stochastic PTNOs with
bi-stable phases emulating artificial Ising spins. We demonstrate the ability
of the stochastic PTNO-CTDS to progressively find more optimal solution by
increasing the strength of the injection-locking signal - akin to performing
classical annealing. We demonstrate in silico that the PTNO-CTDS prototype
solves a benchmark non-deterministic polynomial time (NP)-hard Max-Cut problem
with high probability of success. Using experimentally calibrated numerical
simulations and incorporating non-idealities, we investigate the performance of
our Ising Hamiltonian solver on dense Max-Cut problems with increasing graph
size. We report a high energy-efficiency of 1.3x10^7 solutions/sec/Watt for
100-node dense Max-cut problems which translates to a 5x improvement over the
recently demonstrated memristor-based Hopfield network and several orders of
magnitude improvement over other candidates such as CPU and GPU, quantum
annealer and photonic Ising solver approaches. Such an energy efficient
hardware exhibiting high solution-throughput/Watt can find applications in
industrial planning and manufacturing, defense and cyber-security,
bioinformatics and drug discovery.
