= Stabilizer code
{title2=$\mathcal C=\{\,|\psi\rangle:g_j|\psi\rangle=|\psi\rangle\ \forall j\,\}$}
{wiki}
A <stabilizer code> is the common $+1$ <eigenspace> of a <stabilizer group> $S$, an <abelian subgroup> of the <Pauli group> excluding $-I$. Choose $r$ independent <stabilizer generators> $g_1,\ldots,g_r$ on $n$ physical <qubits>. They are commuting <Hermitian operators>, each squaring to $I$. The <stabilizer subspace> and its <orthogonal projector> are
$$
\mathcal C=\{\,|\psi\rangle:g_j|\psi\rangle=|\psi\rangle\text{ for }j=1,\ldots,r\,\},\qquad
P=\prod_{j=1}^r\frac{I+g_j}{2}=\frac{1}{|S|}\sum_{g\in S}g.
$$
The <stabilizer-projector formula> gives $\dim\mathcal C=2^{n-r}$, so $k=n-r$ <logical qubits> remain. Independent here means that no nonempty product of the chosen <stabilizer generators> is $I$. Excluding $-I$ is necessary: a vector cannot simultaneously be fixed by $I$ and $-I$ unless it is zero.
For the three-<qubit> <bit-flip repetition code>, take $g_1=Z_1Z_2$ and $g_2=Z_2Z_3$, with $Z_j$ the <Pauli Z gate> on site $j$ and identities elsewhere. The <stabilizer group> is $S=\{I,Z_1Z_2,Z_2Z_3,Z_1Z_3\}$. Both <stabilizer generators> have <eigenvalue> $+1$ exactly on the span of $|000\rangle$ and $|111\rangle$, giving $2^{3-2}=2$ dimensions for one <logical qubit>. A <projective measurement> of these checks distinguishes four <orthogonal> two-dimensional sectors rather than eight individual <computational basis> states.
A <Pauli error> that anticommutes with a <stabilizer generator> changes that check's sign, producing an <error syndrome>. A <Pauli operator> commuting with every check can still act on the <logical qubits>; the <centralizer of a stabilizer group> describes these possibilities. Thus the number of checks determines the encoded dimension, while the <quantum code distance> determines protection against arbitrary local errors. The <five-qubit error correcting code>, for instance, has four <stabilizer generators> and <quantum code distance> three; the three-<qubit> <bit-flip repetition code> has two checks but <quantum code distance> one because a single <phase flip> is unflagged. https://www.preskill.caltech.edu/ph229/notes/chap7.pdf[Preskill, §7.9.1: stabilizers, syndromes and logical operators]{ref}.
Back to article page
