Circuit Lower Bounds for the p-Spin Optimization Problem

We consider the problem of finding a near ground state of a p-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors’ recent work [GJW20], we establish that any poly-size n-output circuit that produces a spin assignment with objective value within a certain constant factor of optimality, must have depth at least log n/(2 log log n) as n grows. This is stronger than the known state of the art bounds of the form Ω(log n/(k(n) log log n)) for similar combinatorial optimization problems, where k(n) depends on the optimality value. For example, for the largest clique problem k(n) corresponds to the square of the size of the clique [Ros10]. At the same time our results are not quite comparable since in our case the circuits are required to produce a solution itself rather than solving the associated decision problem. As in our earlier work [GJW20], the approach is based on the overlap gap property (OGP) exhibited by random p-spin models, but the derivation of the circuit lower bound relies further on standard facts from Fourier analysis on the Boolean cube, in particular the Linial-Mansour-Nisan Theorem. To the best of our knowledge, this is the first instance when methods from spin glass theory have ramifications for circuit complexity.