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

TL;DR

This paper explores mappings between various 2D quantum models, including the cluster state, Wen's model, Ising chain, and toric code, revealing their interrelations and implications for topological quantum computation.

Contribution

It introduces a duality transformation that maps the 2D cluster state to Wen's model and analyzes the physical realization and computational universality of these mappings.

Findings

Mapped 2D cluster state to Wen's model via duality

Connected Wen's model to 1D Ising chains through fermionization

Discussed implications for topological quantum computation

Abstract

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.