  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.
