Lloyd-Slotine nine-mode code[1] 

Description

A \([[9,1,3]]_{\mathbb{R}}\) analog CSS version of Shor's nine-qubit code. The nullifiers for this code are \begin{align} \begin{split} &\hat{x}_1 - \hat{x}_2~, \hat{x}_2 - \hat{x}_3~, \hat{x}_4 - \hat{x}_5~ , \hat{x}_5 - \hat{x}_6~ , \hat{x}_7 - \hat{x}_8, \hat{x}_8 - \hat{x}_9~,\\ &(\hat{p}_1 + \hat{p}_2 + \hat{p}_3) - (\hat{p}_4 + \hat{p}_5 + \hat{p}_6)~,\\ &(\hat{p}_4 + \hat{p}_5 + \hat{p}_6) - (\hat{p}_7 +\hat{p}_8 + \hat{p}_9)~. \end{split} \tag*{(1)}\end{align} Logical mode operators are generated by \begin{align} \begin{split} \bar q &=& \hat{q}_1 + \hat{q}_4 + \hat{q}_7~, \\ \bar p &=& \hat{p}_1 + \hat{p}_2 + \hat{p}_3~. \end{split} \tag*{(2)}\end{align}

Realizations

Optical network by the Furusawa group [2].

Parents

Cousin

References

[1]
S. Lloyd and J.-J. E. Slotine, “Analog Quantum Error Correction”, Physical Review Letters 80, 4088 (1998) arXiv:quant-ph/9711021 DOI
[2]
T. Aoki et al., “Quantum error correction beyond qubits”, (2008) arXiv:0811.3734
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Zoo Code ID: lloyd_slotine

Cite as:
“Lloyd-Slotine nine-mode code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/lloyd_slotine
BibTeX:
@incollection{eczoo_lloyd_slotine, title={Lloyd-Slotine nine-mode code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/lloyd_slotine} }
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Cite as:

“Lloyd-Slotine nine-mode code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/lloyd_slotine

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/hyperplane/lloyd_slotine.yml.