\([[9,1,3]]\) Shor code[1]

Nine-qubit CSS code that is the smallest such code to correct a single-qubit error. Logical codewords are \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{2\sqrt{2}}\left(|000\rangle+|111\rangle\right)^{\otimes3}\\ |\overline{1}\rangle&=\frac{1}{2\sqrt{2}}\left(|000\rangle-|111\rangle\right)^{\otimes3}~. \end{split} \tag*{(1)}\end{align} The code works by concatenating each qubit of a phase-flip with a bit-flip repetition code. Therefore, the code can correct both type of errors simultaneously.

Specifically, a state is phase-flip error-corrected by a three-qubit phase-flip repetition code, with stabilizer generators \(X_0 X_1I_2\) and \(X_0I_1X_2\) in \(X\) basis, where the subscript represents the qubit index. Each logical qubit is encoded using \begin{align} \label{eq:phase-flip} \begin{split} |\overline{0}\rangle &= |+_0+_1+_2\rangle \\ |\overline{1}\rangle &= |-_0-_1-_2\rangle . \end{split} \tag*{(2)}\end{align} Then, each physical qubit used in (2) is further encoded in the three-qubit bit-flip repetition code, \begin{align} |\pm _j \rangle = \frac{1}{\sqrt{2}}( |0_{j0}0_{j1}0_{j2}\rangle \pm |1_{j0}1_{j1}1_{j2}\rangle), \tag*{(3)}\end{align} each with bit-flip error stabilizer generators \(Z_{j0}Z_{j1}I_{j2}\) and \(Z_{j0}I_{j1}Z_{j2} \) with \(j=0,1,2\). Notice now the phase-flip error stabilizer generator is extended as \(X_j = X_{j0}X_{j1}X_{j2}\). As a result, the stabilizer generators with the qubit index flattened are \begin{align} \begin{split} Z_{j0}Z_{j1}I_{j2} &\rightarrow \{Z_0Z_1, Z_3Z_4, Z_6Z_7\} \\ Z_{j0}I_{j1}Z_{j2} &\rightarrow \{Z_0Z_2, Z_3Z_5, Z_6Z_8\} \\ X_0 X_1I_2 &\rightarrow \{X_0X_1X_2X_3X_4X_5\}\\ X_0 I_1X_2 &\rightarrow \{X_0X_1X_2X_6X_7X_8\}. \end{split} \tag*{(4)}\end{align}