\([[4,2,2]]_{G}\) 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}
See Ref. [3] for a \([[4,1,2]]_{\mathbb{Z}_q}\) subcode.
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 bit-flip with a phase-flip group-based repetition code for that group.
- 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 bit-flip with a phase-flip group-based repetition code for that group.
Primary Hierarchy
Parents
The four group-qudit code is the smallest quantum double code.
The four group-qudit code is \(G\)-covariant.
\([[4,2,2]]_{G}\) four group-qudit code
Children
The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}\).
The four group-qudit code reduces to the four-rotor code for \(G= \mathbb{Z}_2\).
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, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [3]
- S. Dutta, D. Biswas, and P. Mandayam, “Noise-adapted qudit codes for amplitude-damping noise”, Physical Review A 111, (2025) arXiv:2406.02444 DOI
Page edit log
- Victor V. Albert (2023-11-05) — most recent
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