 Shor code

The nine-qubit Shor code encodes one logical qubit with basis vectors $|0_L⟩=|G_{+}{⟩}^{\otimes 3}$ and $|1_L⟩=|G_{-}{⟩}^{\otimes 3}$, where $|G_{\pm }⟩=(|000⟩\pm |111⟩)/\sqrt{2}$. Six adjacent $Z$-pair stabilizer generators locate a single bit flip within a block, while $X_1\cdots X_6$ and $X_4\cdots X_9$ diagnose which block has had its relative phase changed. Together they correct every single-qubit Pauli operator, hence every single-qubit error by the Pauli expansion of a quantum error.

A Pauli Z gate on qubit four has phase-check eigenvalues $(-1,-1)$ and leaves all six $Z$-pair checks positive. Applying $Z_4$ restores the state. The same phase syndrome arises from $Z_5$ or $Z_6$; these have the same action on the code since their pairwise products are stabilizers. This is a degeneracy, not a failure of correction.