Description
An analog stabilizer version of Shor's nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode.
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}
Decoding
Realizations
Parents
- Analog stabilizer code
- Group-based QPC — The \([[9,1,3]]_{G}\) group-based QPC reduces to the \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code for \(G=\mathbb{R}\).
- Small-distance block quantum code
Cousin
- \([[9,1,3]]\) Shor code — The Lloyd-Slotine nine-mode code is a bosonic analogue of Shor's code.
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]
- S. L. Braunstein, “Error Correction for Continuous Quantum Variables”, Physical Review Letters 80, 4084 (1998) arXiv:quant-ph/9711049 DOI
- [3]
- T. Aoki, G. Takahashi, T. Kajiya, J. Yoshikawa, S. L. Braunstein, P. van Loock, and A. Furusawa, “Quantum error correction beyond qubits”, (2008) arXiv:0811.3734
Page edit log
- Victor V. Albert (2022-06-22) — most recent
Cite as:
“\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/lloyd_slotine