Generalized color code[1] 

Description

Member of a family of nonabelian topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)).

Parent

  • Twisted quantum double (TQD) code — The anyon theory corresponding to a generalized color code is a trivial-cocycle TQD associated with the group \(G \times G/[G,G]\), where \(G\) is any finite group.

Child

  • Color code — The generalized color code for \(G=\mathbb{Z}_2\) reduces to the color code.

Cousins

  • Group GKP code — Generalized color-code Hamiltonians should be expressable in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [2; Sec. 3.3].
  • Quantum-double code — Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the abelianization of \(G\).

References

[1]
C. G. Brell, “Generalized color codes supporting non-Abelian anyons”, Physical Review A 91, (2015) arXiv:1408.6238 DOI
[2]
V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
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Zoo Code ID: generalized_color

Cite as:
“Generalized color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_color
BibTeX:
@incollection{eczoo_generalized_color,
  title={Generalized color code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/generalized_color}
}
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Cite as:

“Generalized color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/generalized_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/groups/topological/generalized_color.yml.