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

TL;DR

This paper explores advanced tensor network contraction techniques to efficiently approximate low-energy states of rugged spin glasses on periodic lattices, surpassing previous planar limitations and achieving high accuracy with low bond dimensions.

Contribution

It introduces hyperoptimized tensor network contraction methods to extend the applicability to non-planar spin glasses, demonstrating effective results on 2D and 3D lattice instances.

Findings

Quadratic time complexity at fixed bond dimension

Approximate solutions within 1-10% of optimal energy

Method effective on large, rugged spin glass instances

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 relative energy error in the most rugged instance class remains at ~1% (square lattice) or ~10% (cubic lattice) of optimal. In less rugged instances the solution is always optimal for the square lattice and either optimal or within ~1% for the cubic lattice. These results suggest that further development of optimization methods based on tensor-network representations of spin glass partition functions may be fruitful, especially given that such methods are not limited to the Ising (i.e., binary) or two-body (i.e., quadratic) settings.