\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code[1] 

Description

Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code using properties of the multiplicative group \(\mathbb{Z}_q\).

Protection

Protects against any quantum error arising from any one of the nine quantum registers.

Encoding

Generalized CNOT, Toffoli, and quantum Fourier transform gates.

Parents

Child

  • \([[9,1,3]]\) Shor code — The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code for \(q=2\) reduces to the \([[9,1,3]]\) Shor code.

Cousins

  • Projective-plane surface code — The qudit Shor code is a small qudit surface code on a Möbius strip with smooth boundary, which is obtained from removing a face of the tesselation of the projective plane \(\mathbb{R}P^2\) [2; Fig. 4].
  • Group-based quantum repetition code — The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code is a concatenation of a two-group-qudit bit-flip with a two-qubit phase-flip group repetition code for \(G=\mathbb{Z}_q\).
  • Concatenated quantum code — The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code is a concatenation of a two-group-qudit bit-flip with a two-qubit phase-flip group repetition code for \(G=\mathbb{Z}_q\).

References

[1]
H. F. Chau, “Correcting quantum errors in higher spin systems”, Physical Review A 55, R839 (1997) arXiv:quant-ph/9610023 DOI
[2]
M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: stab_9_1_3

Cite as:
\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_9_1_3
BibTeX:
@incollection{eczoo_stab_9_1_3, title={\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_9_1_3} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/stab_9_1_3

Cite as:

\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_9_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/small/stab_9_1_3.yml.