Kitaev’s quantum double model as an error correcting code

Shawn X. Cui1, Dawei Ding2, Xizhi Han2, Geoffrey Penington2, Daniel Ranard2, Brandon C. Rayhaun2, and Zhou Shangnan2

1Departments of Mathematics, Physics and Astronomy, Purdue University, West Lafayette, IN 47907
2Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305

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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.

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Cited by

[1] Matthew Buican, Linfeng Li, and Rajath Radhakrishnan, "a×b=c in 2+1D TQFT", Quantum 5, 468 (2021).

[2] Bruno Nachtergaele, Robert Sims, and Amanda Young, "Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States", arXiv:2010.15337, Annales Henri Poincaré (2021).

[3] Yang Qiu and Zhenghan Wang, "Ground subspaces of topological phases of matter as error correcting codes", Annals of Physics 422, 168318 (2020).

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