Linear-time classical approximate optimization of cubic-lattice classical spin glasses

Abstract

Demonstrating quantum speedup for approximate optimization of classical spin glasses is of current interest. Such a demonstration must be done with respect to the best-known scaling of classical heuristics at a given optimality gap of a given problem. For cubic-lattice classical Ising spin glasses, recent theoretical and experimental developments open the possibility of showing quantum speedup for approximate optimization with quantum annealing. It is therefore desirable to understand the optimality-gap range over which such a speedup should be searched for. Here we show that on cubic-lattice tile-planting models, classical meta-heuristics that are linear-time by construction can reach optimality gaps at which simulated annealing and parallel tempering exhibit super-linear scaling. This implies that the optimality gaps achieved by linear-time classical meta-heuristics can serve as useful upper bounds for the optimality-gap range over which quantum speedups in approximate optimization should be searched for. We also explain how classical heuristics with fixed scaling that is beyond-cubic can provide upper bounds to optimality-gap ranges for beyond-quadratic quantum speedups in approximate optimization. These results encourage the development of classical heuristics with fixed scaling that achieve optimality gaps as small as possible.