Graph States

Graph states are a beautiful, intuitive family of stabilizer states. Each one is built from a simple graph: vertices become qubits and edges become entangling gates. They unify cluster states for measurement-based computing, many error-correcting codewords, and a whole visual calculus for entanglement.

Definition

Let $G = (V, E)$ be an undirected graph with one vertex per qubit. The graph state $|G\rangle$ is prepared by:

Initializing every qubit in $|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ .
Applying a controlled- $Z$ gate, $\mathrm{CZ}$ , across every edge $(a,b) \in E$ .

In symbols,

|G\rangle = \Bigg(\prod_{(a,b)\in E} \mathrm{CZ}_{ab}\Bigg)\, |+\rangle^{\otimes |V|}.

Because all $\mathrm{CZ}$ gates commute with one another, the order of applying them is irrelevant — only the edge set matters. Each $\mathrm{CZ}$ is Clifford and $|+\rangle$ is a stabilizer state, so every graph state is a stabilizer state.

The graph-state stabilizers

For each vertex $v$ , the graph state has a stabilizer generator

K_v = X_v \prod_{w \in N(v)} Z_w,

where $N(v)$ is the set of neighbors of $v$ . That is: an $X$ on the vertex itself, dressed by a $Z$ on each adjacent qubit. The $n$ generators $\{K_v\}$ commute (you can check $K_u$ and $K_v$ overlap in an even number of anticommuting positions) and are independent, so they fully specify the unique $n$ -qubit graph state. This is the cleanest possible "state = graph" dictionary: read the neighbors off the picture and you have the stabilizers.

Examples

Single edge $0 - 1$ . $|G\rangle = \mathrm{CZ}\,|{+}{+}\rangle$ . This is local-Clifford equivalent to the Bell state: $(H\otimes I)|G\rangle = \tfrac{1}{\sqrt2}(|00\rangle + |11\rangle)$ .
Triangle (complete graph $K_3$ ). Local-Clifford equivalent to the GHZ state — indeed, every GHZ state is a graph state of a star (or equivalently complete) graph up to local Hadamards.
Linear chain (path graph). The 1D cluster state, the resource for one-dimensional measurement-based computation.

Why graph states matter

Measurement-based quantum computing. A large 2D cluster state (graph state of a grid) is a universal resource: computation proceeds by adaptively measuring individual qubits, no two-qubit gates needed after the cluster is built. This is the one-way quantum computer of Raussendorf and Briegel.
Error correction. Many stabilizer codes' codewords are graph states (up to local Cliffords), and the graph picture clarifies their entanglement structure.
Quantum networks. Graph states model entanglement distributed across a network, where edges are shared entanglement links.

Try it

Build the linear graph state for the path $0 - 1 - 2$ : Hadamard every qubit, then apply $\mathrm{CZ}$ on edges $(0,1)$ and $(1,2)$ . The result is the three-qubit 1D cluster state.

Run your code to see the quantum state.