Shor [[9,1,3]]

The Shor nine-qubit code is a [[9,1,3]] CSS stabilizer code obtained by concatenating three-qubit phase- and bit-flip repetition codes. It corrects an arbitrary single-qubit Pauli error and provides an early concrete example of quantum-code degeneracy.

Arrange the nine qubits as three blocks of three. Six weight-two Z checks detect bit flips within blocks, while two weight-six X checks compare block phases; the codewords are coherent superpositions of three GHZ-like blocks.

Distance three means that the minimum nontrivial logical Pauli has weight three, so one unknown single-qubit Pauli error is correctable. Degenerate errors must be grouped by stabilizer coset rather than counted as distinct Pauli strings.

The canonical instance has n=9, k=1, d=3, and rate 1/9. Its eight independent stabilizer constraints include weight-six X checks, so this finite code is not evidence for a bounded-check-weight qLDPC family.

The CSS structure supports useful blockwise implementations of some Clifford operations, but the CSS label alone does not imply a universal transversal gate set. Gate claims must specify the encoded circuit, correction locations, and fault-propagation analysis.

Decoding separates into two three-bit majority votes: intrablock Z checks locate X components and interblock X checks locate Z components. A Y error contributes to both syndromes and must not be treated as an exclusive third case.

A bare ancilla used for a high-weight check can propagate one ancilla fault into several data errors. Fault-tolerant extraction therefore needs verified cat states, flags, or an equivalent controlled schedule; ideal one-round correction is not a circuit-level threshold statement.

Small platforms can prepare the encoded state and inject single-qubit errors to verify the syndrome table. A meaningful experiment must also report two-qubit-gate, measurement, and idle noise plus postselection; nine data qubits are not the full fault-tolerant resource count.

The Shor code is a CSS concatenation of repetition codes and is closely related to the 3-by-3 Bacon-Shor subsystem code through gauge choice or gauge fixing. Their stabilizer weights, measurable gauge sets, and fault paths differ, so equal [[9,1,3]] parameters do not establish code identity.

Distinguish correction of an arbitrary single-qubit error from fault tolerance under a circuit noise model. The [[9,1,3]] statement identifies an exact finite code; thresholds, pseudothresholds, and resource costs require a specified syndrome circuit and decoder.