Construct and work with a graph-state circuit based on a graph

Graph states (also known as cluster states) are a specific class of multi-qubit entangled states that are defined based on graphs. Graph states are foundational to the one-way quantum computing model (measurement-based quantum computing), where quantum computation is achieved through a sequence of adaptive measurements on an initial graph state.

Generate a random graph with 4 vertexes and 5 edges. Each vertex of the graph corresponds to a qubit, and the edges represent entanglement between the qubits:

Now let's find stabilizers, which are simply a set of operators that represent the symmetries of a quantum state. Given the above list of vertices and their adjacencies find the stabilizers, which can be obtained by Pauli-X on vertex and Pauli-Z on adjacencies: