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Small-distance block quantum code

Description

A block quantum code on \(n\) subsystems that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\).

See Refs. [1–3] for small-distance codes.

Cousins

Member of code lists

Primary Hierarchy

Quantum Domain
Quantum code
Approximate operator-algebra QECC
Operator-algebra QECC (OAQECC)
Quantum error-correcting code (QECC)
Block quantum code
Parents
Block quantum code
Small-distance block quantum code
Children
Kitaev current-mirror qubit code
Three-rotor code
Zero-pi qubit code
Five-rotor code
\([[10,1,4]]_{G}\) tenfold code
\([[4,2,2]]_{G}\) four group-qudit code
\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code
\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code
Perfect quantum code
All non-trivial perfect codes have distance three.
\(((5+2r,3\times 2^{2r+1},2))\) Rains code
Smolin-Smith-Wehner (SSW) code
\(((9,2,3))\) Ruskai code
\(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code
\(((10,24,3))\) qubit code
The \(((10,24,3))\) qubit code can be combined to from an infinite family of distance-three qubit codes whose logical dimension is \(50\%\) larger than that of the optimal stabilizer code [5].
Numerically optimized four-qubit AD code
\(((8,1,3))\) Plenio-Vedral-Knight CE code
\(((9,12,3))\) qubit code
The \(((9,12,3))\) qubit code can be combined to from an infinite family of distance-three qubit codes whose logical dimension is \(50\%\) larger than that of the optimal stabilizer code [5].
Small-distance qubit stabilizer code
\(((3,6,2))_{\mathbb{Z}_6}\) Euler code
\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
\([[11,1,5]]_3\) qutrit Golay code
Three-qutrit code
\([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
\([[9,1,5]]_3\) quantum Glynn code
\([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code
\([[9m-k,k,2]]_3\) triorthogonal code
\(((n,1+n(q-1),2))_q\) union stabilizer code
\([[5,1,3]]_q\) Galois-qudit code
\([[6,2,3]]_{q}\) code
\([[7,3,3]]_{q}\) code
Twisted \(1\)-group code
All twisted \(1\)-group codes have a distance \(d \geq 2\).

References

[1]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[2]
D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh, “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
[3]
W.-T. Yen and L.-Y. Hsu, “Optimal Nonadditive Quantum Error-Detecting Code”, (2009) arXiv:0901.1353
[4]
E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
[5]
S. Yu, Q. Chen, and C. H. Oh, “Two infinite families of nonadditive quantum error-correcting codes”, (2009) arXiv:0901.1935
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Zoo Code ID: small_distance_quantum

Cite as:
“Small-distance block quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/small_distance_quantum
BibTeX:
@incollection{eczoo_small_distance_quantum, title={Small-distance block quantum code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/small_distance_quantum} }
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Cite as:

“Small-distance block quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/small_distance_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/small_distance_quantum.yml.