Graph state

A graph state assigns one qubit to each vertex of an undirected graph. Begin with all qubits in $|+⟩$ and apply a Controlled-Z gate for every edge.

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

The Controlled-Z gates commute, so their ordering does not change the result. On a computational-basis state $|x⟩$, each occupied edge contributes $(-1{)}^{x_a{x}_b}$; the amplitudes retain equal magnitude but acquire graph-dependent phases. An isolated vertex remains in the product state $|+⟩$, while edges can create entanglement. ^[ref]

Let $1$ be a leaf with neighbour $2$ and let the graph on the other vertices be generated by the Controlled-Z gate product $E_{\mathrm{rest}}$. If the leaf has Pauli Z gate frame $Z_1^r$, measuring it at angle $\alpha$ with result $s$ applies one-bit teleportation and leavesAny known Pauli Z gate frames on other vertices remain as additional factors before those vertices' input states. The identity $J(\alpha )Z^r=X^{r}J(\alpha )$ combines the leaf byproduct with the measurement result.

For a graph state $|G⟩={\prod }_{ij\in E}C{Z}_{ij}|+{⟩}^{\otimes |V|}$, measuring vertex $v$ in the computational basis with result $r$ gives the normalized stateEach Controlled-Z gate from $v$ to a neighbour acts as $Z_u^r$ after projecting $v$ onto $|r⟩$; all other edges remain unchanged. The projection contributes $1/\sqrt{2}$, so either result has probability $1/2$. For a four-cycle, deleting one vertex leaves a path and adds byproducts on its two endpoints.

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