Completeness of the classical 2D Ising model and universal quantum computation

TL;DR

This paper demonstrates that the 2D Ising model can represent the partition functions of any classical q-state spin model, establishing its universality and completeness through a connection with measurement-based quantum computation.

Contribution

It proves the completeness of the 2D Ising model for representing all classical spin models and provides a constructive mapping for Ising and Potts models.

Findings

Partition function of any classical q-state model can be expressed as a 2D Ising model instance.

Polynomial overhead in system size for Ising and Potts models.

Connection established between classical spin models and measurement-based quantum computation.

Abstract

We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins w.r.t the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.