# Computational hardness of spin-glass problems with tile-planted solutions

## TL;DR

This paper studies the computational difficulty of spin-glass problems on a lattice with planted solutions, revealing tunable hardness and phase transitions linked to magnetic ordering, using advanced Monte Carlo methods.

## Contribution

It introduces a scalable method for planting solutions in spin-glass problems and analyzes the resulting hardness landscape with multiple phase transitions.

## Key findings

- Problems exhibit a wide range of tunable hardness.
- Multiple hardness phase transitions are observed.
- Harder samples show magnetic ordering transitions.

## Abstract

We investigate the computational hardness of spin-glass instances on a square lattice, generated via a recently introduced tunable and scalable approach for planting solutions. The method relies on partitioning the problem graph into edge-disjoint subgraphs, and planting frustrated, elementary subproblems that share a common local ground state, which guarantees that the ground state of the entire problem is known a priori. Using population annealing Monte Carlo, we compare the typical hardness of problem classes over a large region of the multi-dimensional tuning parameter space. Our results show that the problems have a wide range of tunable hardness. Moreover, we observe multiple transitions in the hardness phase space, which we further corroborate using simulated annealing and simulated quantum annealing. By investigating thermodynamic properties of these planted systems, we demonstrate that the harder samples undergo magnetic ordering transitions which are also ultimately responsible for the observed hardness transitions on changing the sample composition.