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
Page edit log
- Sarah Meng Li (2022-02-21) — most recent
- Victor V. Albert (2022-02-21)
- Victor V. Albert (2023-01-14)
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