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\([[9,1,3]]_{\mathbb{Z}_q}\) 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.

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 \(G=\mathbb{Z}_q\).

Member of code lists

Primary Hierarchy

Quantum Domain
Modular-qudit Kingdom
Modular-qudit codeQECC Quantum
Modular-qudit USt code
Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum
Abelian TQD stabilizer codeLattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Abelian quantum-double stabilizer codeTopological Hamiltonian-based QECC Quantum
Modular-qudit surface codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
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 QPCConcatenated quantum QECC Quantum
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 codeQECC Quantum
\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
Children
\([[9,1,3]]\) Shor code
The \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code for \(q=2\) reduces to the \([[9,1,3]]\) Shor code.

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

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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} }
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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.