\([[4,2,2]]_{G}\) four group-qudit code[1][2; Sec. VIII] 

Description

\([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits.

For elements \(g_1 ,g_2\) of any finite group \(G\), a set of codewords is \begin{align} |\overline{g_{1},g_{2}}\rangle=\frac{1}{\sqrt{|G|}}\sum_{g\in G}|g,gg_{1},gg_{2},gg_{1}g_{2}\rangle~. \tag*{(1)}\end{align}

Parents

Children

  • Four-rotor code — The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}\).
  • \([[4,2,2]]\) CSS code — The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}_2\).

Cousins

  • \([[2m,2m-2,2]]\) error-detecting code — The four group-qudit code can be extended to the \([[2m,2m-2,2]]_{G}\) group-qudit code [2; Sec. VIII]. The latter reduces to the \([[2m,2m-2,2]]\) error-detecting code for \(G=\mathbb{Z}_2\).
  • Group-based QPC — The \(|\overline{g_1=1,g_2}\rangle\) \([[4,1,2]]_{G}\) subcode is the smallest group-based QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip group-based repetition code.
  • Concatenated quantum code — The \(|\overline{g_1=1,g_2}\rangle\) \([[4,1,2]]_{G}\) subcode is the smallest group-based QPC, i.e., a concatenation of a two-qubit bit-flip with a two-qubit phase-flip group-based repetition code for that group.

References

[1]
A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
[2]
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
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Zoo Code ID: group_4_2_2

Cite as:
\([[4,2,2]]_{G}\) four group-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/group_4_2_2
BibTeX:
@incollection{eczoo_group_4_2_2, title={\([[4,2,2]]_{G}\) four group-qudit code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_4_2_2} }
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Cite as:

\([[4,2,2]]_{G}\) four group-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/group_4_2_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/small/group_4_2_2.yml.