Mapping classical spin models to the graph state formalism

Abstract

In this talk we discuss how large classes of classical spin models, such as the Ising and Potts models on arbitrary lattices, can be mapped to the graph state formalism. In particular, we show how the partition function of a spin model can be written as the overlap between a graph state and a complete product state. Here the graph state encodes the interaction pattern of the spin model---i.e., the lattice on which the model is defined---whereas the product state depends only on the couplings of the model, i.e., the interaction strengths. As main examples, we find that the 1D Ising model corresponds to the 1D cluster state, the 2D Ising model without external field is mapped to Kitaev's toric code state, and the 2D Ising model with external field corresponds to the 2D cluster state---but the mappings are completely general in that arbitrary graphs, and also q-state models can be treated. These mappings allow one to make connections between concepts in (classical) statistical mechanics and quantum information theory and to obtain a cross-fertilization between both fields. As a main application, we will prove that the classical Ising model on a 2D square lattice (with external field) is a "complete model", in the sense that the partition function of any other spin model---i.e., for q-state spins on arbitrary lattices---can be obtained as a special instance of the (q=2) 2D Ising partition function with suitably tuned (complex) couplings. This result is obtained by invoking the above mappings from spin models to graph states, and the property that the 2D cluster states are universal resource states for one-way quantum computation. Joint work with Wolfgang Duer and Hans Briegel, see PRL/ 98 117207 (2007)/ and quant-ph/0708.2275. For related work, see also S. Bravyi and R. Raussendorf, quant-ph/0610162.