Abstract
In this paper, we prove, extend and review possible mappings between the two-dimensional (2D) cluster state, Wen's model, the 2D Ising chain and Kitaev's toric code model. We introduce a 2D duality transformation to map the 2D lattice cluster state into the topologically ordered Wen model. Then, we investigate 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 1D Ising interactions. Considering the boundary terms with this mapping then reveals how the Ising chains interact with one another. The duality of these models can be taken as a starting point to address questions as to how their gate operations in different quantum computational models can be related to each other.
Highlights
The connection between Wen’s model and Kitaev’s toric code model has been studied by Nussinov and Ortiz [7]. We investigate this transformation for periodic boundary conditions, and look for the necessary conditions for the transformation to be faithful and when the mapping is mismatched
We have reviewed and proven the relations between four 1D and 2D lattice spin models, namely the 2D cluster state model, Wen’s model, Kitaev’s toric code and the 1D Ising model
Perhaps the most interesting way is the comparison between the 2D cluster state and the Kitaev model. Both models can be used in the possible realization of quantum computation by making use of two very different ground state properties of the system, i.e. the entanglement and the topological structure
Summary
Consider a D-dimensional square lattice and associate with each site of the lattice a spin-half particle. They cannot be performed simultaneously because they do not commute with one another on pairs of nearest neighbour sites With this in mind, and assuming that each of the controlled-NOT gates requires only one unit of time to be performed, and all of the Hadamard gates can be performed simultaneously (as they all occur on different sites), and each of the Ui operators can be performed simultaneously, the time it takes to transform a cluster state into a topologically ordered system scales linearly with the length of the boundary of the lattice. This matches the rate at which topological order is generated in [6], which is recognized in [19]
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