\(((9,2,3))\) Ruskai code[1] 

Description

Nine-qubit permutation-invariant code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes.

In terms of Dicke states, the codewords are \begin{align} \begin{split} |0_{L}\rangle&\propto|D_{0}^{9}\rangle+\sqrt{3}|D_{6}^{9}\rangle\\ |1_{L}\rangle&\propto\sqrt{3}|D_{3}^{9}\rangle+|D_{9}^{9}\rangle~. \end{split} \tag*{(1)}\end{align}

Protection

Protects against all single-qubit errors as well as two-qubit errors arising from exchange processes.

Parents

Cousin

  • \([[9,1,3]]\) Shor code — The \(((9,2,3))\) Ruskai code results from projecting the Shor code into the permutation-invariant qubit subspace [1].

References

[1]
M. B. Ruskai, “Pauli Exchange Errors in Quantum Computation”, Physical Review Letters 85, 194 (2000) arXiv:quant-ph/9906114 DOI
[2]
Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
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Zoo Code ID: ruskai

Cite as:
\(((9,2,3))\) Ruskai code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ruskai
BibTeX:
@incollection{eczoo_ruskai, title={\(((9,2,3))\) Ruskai code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/ruskai} }
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Cite as:

\(((9,2,3))\) Ruskai code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/ruskai

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/transversal/ruskai.yml.