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

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.

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References in corpus (8)

• Non-Abelian Anyons and Topological Quantum Computation
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Taming multiparticle entanglement
• Quantum phase transition between cluster and antiferromagnetic states
• Majorana Fermions, Exact Mappings between Classical and Topological Orders
• Fundamentals of universality in one-way quantum computation
• Topological phases and quantum computation
• Entanglement and local information access for graph states

Cited by in corpus (17)

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• Quantum phase transition between cluster and antiferromagnetic states
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• Topological order in 1D Cluster state protected by symmetry
• Universal fault-tolerant measurement-based quantum computation
• Generalized Cluster States Based on Finite Groups
• Benchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model
• The XYZ${}^2$ hexagonal stabilizer code
• Optimal local unitary encoding circuits for the surface code
• An optimal dissipative encoder for the toric code
• Scaling of geometric phase versus band structure in cluster-Ising models
• Fault-tolerant gates via homological product codes
• Dual quantum-correlation paradigms exhibit opposite statistical-mechanical properties
• Non-perturbative Floquet engineering of the toric-code Hamiltonian and its ground state
• Practical quantum error correction with the XZZX code and Kerr-cat qubits
• Duality in spin systems via the $SU(4)$ algebra
• Free fermion representation of the topological surface code