Stabilizers and Syndromes

A stabilizer is a parity check — a measurement that asks "are these qubits the same or different?" without ever asking what value either one holds. The answers from all the checks form a bit pattern called the syndrome, and each syndrome points to exactly one correction. That is how a code can locate an error without collapsing the superposition it is protecting.

How can you check a qubit without reading it?

Error correction seems to demand the impossible. To fix an error you must look for it, but measuring a qubit collapses its superposition — the very thing the code is protecting. A superposition is a state that is partly |0⟩ and partly |1⟩, with weights called amplitudes; reading the qubit forces it to one value and destroys the weights.

The escape is to measure relational information only. A parity check on two qubits asks one question: "are you the same or different?" — the XOR of the two values — and nothing more. In quantum terms this is measuring the operator Z⊗Z (the symbol ⊗ just means the two single-qubit Zs are bundled into one joint question asked of the pair; the answer comes back as +1 or −1 rather than as either qubit's own value): it answers +1 for |00⟩ and |11⟩ ("same") and −1 for |01⟩ and |10⟩ ("different").

Here is why that is safe. Take the entangled state α|00⟩ + β|11⟩ (α and β are the amplitudes). Both branches, |00⟩ and |11⟩, give the same answer to the parity question: "same." Since every branch agrees, the measurement learns nothing about α and β, and the superposition survives untouched. The check would only disturb a state whose branches disagree about the answer — which is exactly what an error creates. Errors get caught; data passes through unread.