Classical spin systems and the quantum stabilizer formalism: General mappings and applications

Abstract

We present general mappings between classical spin systems and quantum physics. More precisely, we show how to express partition functions and correlation functions of arbitrary classical spin models as inner products between quantum stabilizer states and product states, thereby generalizing mappings for some specific models established in our previous work [M. Van den Nest et al., Phys. Rev. Lett. 98, 117207 (2007)]. For Ising- and Potts-type models with and without external magnetic field, we show how the entanglement features of the corresponding stabilizer states are related to the interaction pattern of the classical model, while the choice of product states encodes the details of the interaction. These mappings establish a link between the fields of classical statistical mechanics and quantum information theory, which we utilize to transfer techniques and methods developed in one field to gain insight into the other. For example, we use quantum information techniques to recover well known duality relations and local symmetries of classical models in a simple way and provide new classical simulation methods to simulate certain types of classical spin models. We show that in this way all inhomogeneous models of q-dimensional spins with pairwise interaction pattern specified by a graph of bounded tree width can be simulated efficiently. Finally, we show relations between classical spin models and measurement-based quantum computation.