Kitaev's quantum double model as an error correcting code

Abstract

Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes -- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the "log dim R" term is included in the definition of entanglement entropy.

added a subsection and two appendices on topological entanglement entropy; fixed some typos; Accepted in Quantum

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• Renormalization group and approximate error correction
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• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Anyon Quantum Dimensions from an Arbitrary Ground State Wave Function
• Local topological order and boundary algebras
• The category of anyon sectors for non-abelian quantum double models
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints