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. Ideal codewords are, for \(g_1 ,g_2 \in G\), \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
- Quantum-double code — The four group-qudit code is the smallest quantum double code.
- Covariant code — The four group-qudit code is \(G\)-covariant.
- Small-distance block quantum code
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 \(G= \mathbb{Z}_2\).
Cousin
- \([[2m,2m-2,2]]\) 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 \(G=\mathbb{Z}_2\).
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
Page edit log
- Victor V. Albert (2023-11-05) — most recent
Cite as:
“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/tree/main/codes/quantum/groups/group_4_2_2.yml.