Hyperoptimized approximate contraction of tensor networks for rugged-energy-landscape spin glasses on periodic square and cubic lattices

Abstract

Obtaining the low-energy configurations of spin glasses that have rugged energy landscapes is of direct relevance to combinatorial optimization and fundamental science. Search-based heuristics have difficulty with this task due to the existence of many local minima that are far from optimal. The work of [M. M. Rams et al., Phys. Rev. E 104, 025308 (2021)] demonstrates an alternative that can bypass this issue for spin glasses with planar or quasi-planar geometry: sampling the Boltzmann distribution via approximate contractions of tensor networks. The computational complexity of this approach is due only to the complexity of contracting the network, and is therefore independent of landscape ruggedness. Here we initiate an investigation of how to take this approach beyond (quasi-)planar geometry by utilizing hyperoptimized approximate contraction of tensor networks [J. Gray and G. K.-L. Chan, Phys. Rev. X 14, 011009 (2024)]. We perform tests on the periodic square- and cubic-lattice, planted-solution Ising spin glasses generated with tile planting [F. Hamze et al., Phys. Rev. E 97, 043303 (2018)] for up to 2304 (square lattice) and 216 (cubic lattice) spins. For a fixed bond dimension, the time complexity is quadratic. With a bond dimension of only four, over the tested system sizes the average solution quality in the most rugged instance class remains at ~1% (square lattice) or ~10% (cubic lattice) of optimal. These results encourage further investigation of tensor network contraction for rugged-energy-landscape spin-glass problems, especially given that this approach is not limited to the Ising (i.e., binary) or two-body (i.e., quadratic) settings.