 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]

A **graph state** is a special type of quantum state associated with a certain graph in quantum information theory. Graph states are fundamental in the context of quantum computing and quantum information processing, particularly in the study of quantum entanglement. Here's a more detailed explanation: 1. **Graph Representation**: A graph \( G \) is defined by a set of vertices (or nodes) \( V \) and edges \( E \) that connect pairs of vertices.