From Near to Eternity: Spin-glass planting, tiling puzzles, and constraint satisfaction problems

TL;DR

This paper introduces a method to generate complex Ising Hamiltonians with known ground states by decomposing the graph into subgraphs, enabling tunable difficulty levels for solving constraint satisfaction problems like tiling puzzles.

Contribution

The authors propose a novel graph decomposition technique to create Ising models with controllable complexity and known solutions, bridging spin-glass models and tiling puzzles.

Findings

Hamiltonians span a wide range of computational difficulty.

Subproblem decomposition allows precise control over problem complexity.

Experimental results confirm the tunability of problem hardness.

Abstract

We present a methodology for generating Ising Hamiltonians of tunable complexity and with a priori known ground states based on a decomposition of the model graph into edge-disjoint subgraphs. The idea is illustrated with a spin-glass model defined on a cubic lattice, where subproblems, whose couplers are restricted to the two values {-1,+1}, are specified on unit cubes and are parametrized by their local degeneracy. The construction is shown to be equivalent to a type of three-dimensional constraint satisfaction problem known as the tiling puzzle. By varying the proportions of subproblem types, the Hamiltonian can span a dramatic range of typical computational complexity, from fairly easy to many orders of magnitude more difficult than prototypical bimodal and Gaussian spin glasses in three space dimensions. We corroborate this behavior via experiments with different algorithms and discuss generalizations and extensions to different types of graphs.