Steane [7,1,3] Code

Steane code: Encodes 1 logical qubit in 7 physical qubits with distance 3 — corrects any single-qubit error. Based on the classical [7,4,3] Hamming code (CSS construction: use two copies of the same classical code for X- and Z-errors independently). Syndrome measurement: 6 stabilizer generators (H-matrix rows) measure parity of qubit subsets without disturbing the logical state. The 6-bit syndrome uniquely identifies which of 7 qubits is faulty (or no error if syndrome = 000000). Error → syndrome → Pauli correction → perfect recovery. No measurement of the logical qubit is ever needed!