= Shor code
{c}
{title2=$[[9,1,3]]$}
The nine-qubit Shor code encodes one logical <qubit> with basis vectors $|0_L\rangle=|G_+\rangle^{\otimes3}$ and $|1_L\rangle=|G_-\rangle^{\otimes3}$, where $|G_\pm\rangle=(|000\rangle\pm|111\rangle)/\sqrt2$. 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.
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