Stabilizer formalism and syndromes

A stabilizer is an Abelian subgroup of the n-qubit Pauli group that excludes minus identity. Choose r independent Hermitian generators. Their common positive eigenspace has dimension $2^{n-r}$ and stores $k=n-r$ logical qubits. Independence matters: multiplying generators may produce additional checks without imposing additional constraints. Including minus identity would demand that a nonzero state equal its own negative and would leave no code space at all.

Check compatibility is an operator condition, not a statement about whether their symbols look different. Two Pauli strings acquire a minus sign for each site where their nonidentity single-qubit factors anticommute. An even number of such sites gives overall commutation. Thus overlapping X and Z checks can be compatible, provided the overlap has even parity. This rule allows simultaneous syndrome measurements in principle, though the circuits that implement those measurements still require a schedule and a fault model.

Represent an error by a Pauli E, ignoring its physically irrelevant overall phase. If E anticommutes with a check, the error changes that check eigenvalue from plus one to minus one. Encode a negative outcome by syndrome bit one. The syndrome is therefore a list of commutation signs, independent of the logical amplitudes. It identifies an error sector and may leave several possible physical errors within that sector.

Binary notation stores a Pauli as two length-n vectors x and z: each site contains an X factor, a Z factor, both for Y up to phase, or neither. The commutation sign of two strings is their binary symplectic product. All additions and inner products here use arithmetic modulo two. Ordinary real dot products would give the wrong criterion. Binary notation speeds syndrome calculations but suppresses phases that must be restored when calculating actual gate products or state amplitudes.

The Pauli normalizer consists of strings commuting with every stabilizer. Some are stabilizers themselves and act as identity on the code. The others implement nontrivial logical operations while producing zero syndrome. Logical X and Z representatives for the same logical qubit must anticommute, even though each commutes with all checks. Multiplying a representative by a stabilizer changes its physical support without changing its code action.

Distance is the minimum weight of a nontrivial Pauli logical operator. With perfect syndrome measurements, a distance-d code corrects arbitrary unknown Pauli errors of weight at most the integer part of half of d minus one. It detects every harmful Pauli below distance, while stabilizer elements are harmless. Check weight is a different quantity: high-weight checks do not establish high distance. Faulty measurements and propagating gate faults need a circuit-level analysis beyond this data-error statement.