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.
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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.
The Controlled-Z gates commute, so their ordering does not change the result. On a computational-basis state , each occupied edge contributes ; the amplitudes retain equal magnitude but acquire graph-dependent phases. An isolated vertex remains in the product state , while edges can create entanglement. [ref]
Qiskit quantum gate documentation by IBM. IBM Quantum, GraphState: preparation by plus states and one controlled-Z per edge