  Ising solvers offer a promising physics-based approach to tackle the
challenging class of combinatorial optimization problems. However, typical
solvers operate in a quadratic energy space, having only pair-wise coupling
elements which already dominate area and energy. We show that such
quadratization can cause severe problems: increased dimensionality, a rugged
search landscape, and misalignment with the original objective function. Here,
we design and quantify a higher-order Hopfield optimization solver, with 28nm
CMOS technology and memristive couplings for lower area and energy
computations. We combine algorithmic and circuit analysis to show quantitative
advantages over quadratic Ising Machines (IM)s, yielding 48x and 72x reduction
in time-to-solution (TTS) and energy-to-solution (ETS) respectively for Boolean
satisfiability problems of 150 variables, with favorable scaling.
