Cluster States

A cluster state is the entangled resource that powers the one-way model. It is built by a recipe so simple it can be stated in one line — initialize every qubit in $|+\rangle$ and apply a controlled-$Z$ across every edge of a graph — yet the resulting state carries exactly the entanglement structure that single-qubit measurements can later sculpt into any quantum algorithm.

The construction

Fix a graph $G = (V, E)$: a set of vertices $V$, one qubit each, and a set of edges $E$. The cluster state $|C_G\rangle$ is defined by

Two facts make this well defined regardless of the order you build it:

• All controlled-$Z$ gates commute with one another, since each is diagonal in the computational basis. The product over edges is unambiguous.
• $CZ$ is symmetric in its two qubits, $CZ_{ab} = CZ_{ba}$, so an edge has no direction.

The controlled-$Z$ gate is diagonal,

flipping the sign of $|11\rangle$ and leaving the other three computational basis states untouched.

The smallest example: $|C_2\rangle$

Take the graph with two vertices and a single edge, $V = \{0, 1\}$, $E = \{(0,1)\}$. Start from

and apply $CZ_{01}$. Only the $|11\rangle$ term picks up a minus sign:

You can check this is maximally entangled: it is local-unitary equivalent to a Bell state. Applying a Hadamard to the second qubit gives $\tfrac{1}{\sqrt2}(|00\rangle + |11\rangle)$. The single minus sign is the entire entanglement content of the two-qubit cluster.

The stabilizer viewpoint

Cluster states are stabilizer states: each is the unique $+1$ eigenstate of a commuting set of Pauli operators. For every vertex $a$,

There is one such stabilizer generator per qubit, the generators commute, and together they pin down $|C_G\rangle$ completely. For $|C_2\rangle$ the two generators are $K_0 = X_0 Z_1$ and $K_1 = Z_0 X_1$. This stabilizer description is the workhorse of the next several lessons: it lets us track what a single-qubit measurement does to the rest of the state without ever writing out an exponentially large state vector.

Larger graphs, richer resources

A linear cluster (a path graph) is a one-dimensional chain; measuring along it teleports a state from one end to the other while applying a programmable single-qubit gate at each step — the subject of the following lessons. A two-dimensional lattice, such as a square grid, adds the cross-links needed for entangling gates between separate logical wires, which is what makes the resource universal. Crucially, the recipe never changes: more qubits in $|+\rangle$, more $CZ$ gates along more edges.

Try it

Construct the two-qubit cluster state $|C_2\rangle = \tfrac12(|00\rangle + |01\rangle + |10\rangle - |11\rangle)$. Put both qubits in $|+\rangle$, then entangle them with a single controlled-$Z$ on the edge $(0,1)$. The grader checks all four amplitudes — note the lone minus sign on $|11\rangle$.

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