Shor nine-qubit error-correction code

Shor's nine-qubit code protects one logical qubit against an arbitrary single-qubit Pauli error by nesting two three-qubit repetition codes in complementary bases. First, $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ is encoded phase-flip-style into $\alpha|{+}{+}{+}\rangle+\beta|{-}{-}{-}\rangle$ using $H$ and CNOTs, then each of those three qubits is bit-flip-encoded via $|0\rangle\to|000\rangle$, $|1\rangle\to|111\rangle$, giving nine physical qubits. Ancilla-assisted stabilizer measurements ($Z_iZ_j$ within each triple, and $X$-type parity across triples) identify which error class occurred — bit flip, phase flip, or both — without collapsing the encoded logical amplitudes $\alpha,\beta$, and a decoder applies the matching Pauli correction. This record documents that stabilizer structure as a code-level reference, matching its own caveat: whether encoding nine noisy physical qubits actually beats one bare qubit depends on the calibrated noise model, since encoding, syndrome extraction, and ancilla errors can outweigh the protection on small noisy devices.