\([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code[1,2]

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}