Stabilizer group

A stabilizer group is an abelian subgroup of the Pauli group that does not contain $-I$. Its simultaneous positive eigenspace is its stabilizer subspace.

A set of stabilizer generators is an independent commuting family of Hermitian Pauli operators whose products form a stabilizer group.

A stabilizer state on $n$ qubits is the unique simultaneous positive eigenstate of a stabilizer group with $n$ independent generators.

Stabilizer-state preparation starts from a computational-basis state and applies a Clifford circuit, producing a stabilizer state.

A stabilizer tableau represents each Pauli generator by a binary row $[x_1\cdots x_n\mid z_1\cdots z_n]$, with one additional bit when its sign is retained. Clifford gates update these rows by arithmetic over ${\mathbb{F}}_2$.

For an undirected graph $G=(V,E)$, the graph state is

The graph-state stabilizer generator at vertex $v$ is $S_v=X_v{\prod }_{u\in N(v)}{Z}_u$. Two such generators commute because adjacent vertices contribute two Pauli anticommutations and nonadjacent vertices contribute none.

Local complementation at a vertex $v$ toggles every edge between neighbors of $v$. It is implemented on graph states, up to global phase, by the local Clifford operator

The stabilizer subspace of a commuting family $S$ of Hermitian Pauli operators isIf there are $r$ independent generators on $n$ qubits, its dimension is $2^{n-r}$.

For a stabilizer group $G=⟨g_1,\dots ,g_l⟩$, the orthogonal projector onto its stabilizer subspace is