Generating topological order from a 2D cluster state using a duality mapping

In this paper we prove, extend and review possible mappings between the two-dimensional Cluster state, Wen's model, the two-dimensional Ising chain and Kitaev's toric code model. We introduce a two-dimensional duality transformation to map the two-dimensional lattice cluster state into the topologically-ordered Wen model. Then, we subsequently investigates how this mapping could be achieved physically, which allows us to discuss the rate at which a topologically ordered system can be achieved. Next, using a lattice fermionization method, Wen's model is mapped into a series of one-dimensional Ising interactions. Considering the boundary terms with this mapping then reveals how the Ising chains interact with one another. The relationships discussed in this paper allow us to consider these models from two different perspectives: From the perspective of condensed matter physics these mappings allow us to learn more about the relation between the ground state properties of the four different models, such as their entanglement or topological structure. On the other hand, we take the duality of these models as a starting point to address questions related to the universality of their ground states for quantum computation.