Cluster and Graph States

GHZ and W states each entangle every qubit with every other qubit, all at once, with no extra structure. Graph states do something different and arguably more useful: they let you choose exactly which qubits are entangled with which, using an ordinary picture — a graph — as the literal blueprint.

The recipe

1. Draw any graph: some dots (vertices), some lines connecting pairs of them (edges).
2. Put one qubit on each vertex, every one starting in the state |+⟩.
3. For every edge in the graph, apply a CZ gate between the two qubits it connects.

That's the entire construction. The resulting entangled state is called a graph state — and a cluster state is just the special case where the graph happens to be a regular lattice (a grid).

Why this matters: measurement-based quantum computing

Build one large graph state in advance, then perform only single-qubit measurements, chosen adaptively based on earlier outcomes — and you can implement any quantum circuit at all. This is "one-way" or measurement-based quantum computing: all the entanglement is prepared up front as one big resource, and the actual "program" is just which basis you measure each qubit in. No further two-qubit gates are ever needed once the graph state exists.

Why cluster states specifically, and why photonics loves them

A 2D cluster state (a square-lattice graph) is provably enough to run any quantum circuit via measurement alone — this was the original Raussendorf–Briegel result that started the entire measurement-based computing field. Cluster/graph states are also the natural fit for photonic quantum computing, where two-qubit gates between photons are notoriously hard to do reliably, but a large entangled "web" of photons can be built up in advance from smaller pieces (a process called fusion) and then consumed entirely through single-photon measurements — exactly the architecture several leading photonic quantum computing efforts are built around.