Stabilizer Measurements and Syndromes
Measuring the stabilizer generators yields a syndrome, a pattern of eigenvalues that reveals errors without disturbing the encoded information.
What a syndrome is
A codeword lies in the +1 eigenspace of every stabilizer generator. When an error E strikes, each generator g either still reads +1, if E commutes with g, or reads -1, if E anticommutes with g. The list of these +-1 outcomes, recorded as bits, is the syndrome. It is a fingerprint of the error class, and crucially it is measured without learning the encoded amplitudes.
The reason the encoded data survives is that the stabilizers commute with the logical operators. Measuring a stabilizer projects onto the +1 or -1 subspace of that stabilizer but does not collapse the logical superposition, which lives orthogonally to that distinction.
From syndrome to correction
Different errors can share a syndrome; the set of all Paulis with a given syndrome forms a coset. A decoder selects the most probable error consistent with the observed syndrome, typically the lowest-weight one, and its inverse is applied. If the guess differs from the true error by a stabilizer, the correction still works, because stabilizers act trivially on codewords. If it differs by a logical operator, a logical error results.
- Syndrome = pattern of stabilizer eigenvalues after an error.
- +1 means the error commutes with that generator; -1 means it anticommutes.
- Measuring stabilizers reveals errors but not the logical state.
- A decoder maps syndrome to a likely correction; success is up to a stabilizer.
Measurement without collapse
In practice a stabilizer is measured indirectly using an ancilla qubit and controlled gates, so the multi-qubit Pauli is read out as a single classical bit. See syndrome extraction circuits for the standard ancilla-coupled construction. Real measurements are themselves noisy, so syndromes are extracted repeatedly and the results are decoded together over space and time, a point developed in measurement errors and repeated measurement.
Continuous errors are digitized by this process: a small rotation is projected by the syndrome measurement onto either no error or a full Pauli, so the decoder only ever faces a discrete problem.