Generalized 2D color code[1] 

Description

Member of a family of non-Abelian 2D topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)).

Decoding

Chromobius, an open-source implementation of the Mobius decoder works for many 2D color codes [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

  • 2D color code — The generalized color code for \(G=\mathbb{Z}_2\) reduces to the 2D 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 [3; 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 \(G\).
  • Modular-qudit color code — The generalized color code for \(G=\mathbb{Z}_q\) reduces to the 2D modular-qudit color code.

References

[1]
C. G. Brell, “Generalized color codes supporting non-Abelian anyons”, Physical Review A 91, (2015) arXiv:1408.6238 DOI
[2]
C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
[3]
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 2D 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 2D 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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“Generalized 2D 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/edit/main/codes/quantum/groups/topological/generalized_color.yml.