\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code[1,2] 

Description

Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [1]; see also [2; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations. A concise expression for a set of codewords can be found in [3; Sec. VI.B].

Protection

Protects against a single error on any one qudit. Detects two-qudit errors.

Encoding

Generalized CNOT, Toffoli, and quantum Fourier transform gates.

Parents

Child

  • Five-qubit perfect code — The \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code for \(q=2\) reduces to the five-qubit perfect code.

Cousins

  • Five-rotor code — The five-rotor code is a rotor analogue of the five-qudit code.
  • Tenfold code — The \([[10,1,4]]_{G}\) Abelian group code for \(G=\mathbb{Z}_q\) is defined using a graph that is closely related to the \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code [4].
  • Braunstein five-mode code — The Braunstein five-mode code is a bosonic analogue of the five-qudit code.

References

[1]
H. F. Chau, “Five quantum register error correction code for higher spin systems”, Physical Review A 56, R1 (1997) arXiv:quant-ph/9702033 DOI
[2]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[3]
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
[4]
D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
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Zoo Code ID: qudit_5_1_3

Cite as:
\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_5_1_3
BibTeX:
@incollection{eczoo_qudit_5_1_3, title={\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_5_1_3} }
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https://errorcorrectionzoo.org/c/qudit_5_1_3

Cite as:

\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_5_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/small/qudit_5_1_3.yml.