The graph-state stabilizer generator at vertex is . Two such generators commute because adjacent vertices contribute two Pauli anticommutations and nonadjacent vertices contribute none.
Local complementation at a vertex toggles every edge between neighbors of . It is implemented on graph states, up to global phase, by the local Clifford operator
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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.