6 CSS structure
A CSS code is one whose stabilizer generators are each purely \(X\)-type or purely \(Z\)-type. The separation is what lets a homological argument apply: the \(X\) and \(Z\) sides become the two ends of a chain complex.
A Pauli group element whose operator part uses only \(I\) and \(X\), with trivial phase.
A Pauli group element whose operator part uses only \(I\) and \(Z\), with trivial phase. A CSS code is one whose stabilizer is generated by elements each of which is \(X\)-type (definition 42) or \(Z\)-type.
If every weight-one Pauli anticommutes with some stabilizer generator, and a weight-two nontrivial logical operator exists, then the code has distance exactly \(2\).
The anticommuting witnesses exclude weight-one elements from the centralizer, so by definition 31 no nontrivial logical operator has weight one; weight zero is the identity, which is a stabilizer. The exhibited weight-two logical then makes the minimum exactly two by definition 39. This single lemma discharges the distance obligation for the whole family of detection codes in chapter 10.
Let a CSS code have \(Z\)-checks and \(X\)-checks supported on the rows of two classical parity-check matrices. If both matrices have nonzero, pairwise distinct columns — both classical codes have distance at least \(3\) — and some nontrivial logical operator of weight \(3\) exists, then the code has distance exactly \(3\).
A weight-one Pauli at qubit \(i\) is \(Z\), detected by any \(X\)-row through \(i\), or carries an \(X\)-component, detected by any \(Z\)-row through \(i\); nonzero columns supply the rows. For a weight-two Pauli on qubits \(i \ne j\), the anticommutation with a check is the sum of the two single-qubit symplectic products it sees: if exactly one qubit carries an \(X\)-component a \(Z\)-row through it detects the pair, and if both (or neither, so both are \(Z\)) do, a \(Z\)-row (resp. \(X\)-row) containing exactly one of \(i, j\) — which exists because the columns are distinct — sees a single \(1\). So no Pauli of weight one or two lies in the centralizer, and the exhibited weight-three logical makes the minimum exactly three by definition 39.