Kitaev's quantum double model as an error correcting code

Shawn X. Cui; Dawei Ding; Xizhi Han; Geoffrey Penington; Daniel Ranard; Brandon C. Rayhaun; Zhou Shangnan

doi:10.22331/q-2020-09-24-331

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

Published:	2020-09-24, volume 4, page 331
Eprint:	arXiv:1908.02829v3
Doi:	https://doi.org/10.22331/q-2020-09-24-331
Citation:	Quantum 4, 331 (2020).

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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.

► BibTeX data

@article{Cui2020kitaevsquantum,
  doi = {10.22331/q-2020-09-24-331},
  url = {https://doi.org/10.22331/q-2020-09-24-331},
  title = {Kitaev's quantum double model as an error correcting code},
  author = {Cui, Shawn X. and Ding, Dawei and Han, Xizhi and Penington, Geoffrey and Ranard, Daniel and Rayhaun, Brandon C. and Shangnan, Zhou},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {331},
  month = sep,
  year = {2020}
}

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