Completeness of the Classical 2D Ising Model and Universal Quantum Computation

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 with respect to 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.

Figure 1

Decorated graph $\tilde{G}$ (right) corresponding to a 2D lattice $G$ (left). Dark (green) dots indicate vertices originating from the vertices $V$ of $G$, while light (red) dots indicate vertices $V_E$ originating from edges $E$ of $G$.