\([[9,1,3]]_{\mathbb{Z}_q}\) 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
- Modular-qudit 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 QPC — The \([[9,1,3]]_{G}\) group-based QPC reduces to the \([[9,1,3]]_{\mathbb{Z}}\) modular-qudit code for \(G=\mathbb{Z}_q\).
- Small-distance block quantum code
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 bit-flip with a 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 bit-flip with a 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
- Sarah Meng Li (2022-02-21) — most recent
- Victor V. Albert (2022-02-21)
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.