Graph states (also known as cluster states) are a specific class of multi-qubit entangled states that are defined based on graphs. Each vertex of the graph corresponds to a qubit, and the edges represent entanglement between the qubits. 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. Using QuantumCircuitOperator[{"Graph",Graph[…]}], one can create a circuit that generates the corresponding quantum graph state.

Generate a random graph with 4 vertexes and 5 edges:

g=RandomGraph[{4,5},VertexLabels->Automatic]

Create the corresponding quantum circuit:

qc=QuantumCircuitOperator[{"Graph",g}]

Circuit diagram:

qc["Diagram"]

Show that the circuit acting on a register state creates the expected quantum cluster/graph state:

ψf=qc[]

ψf==QuantumState[{"Graph",g}]

Show qubits connected via an edge are entangled:

QuantumEntangledQ[ψf,{#1,#}]&@@@EdgeList[g]

Given a vertex of the graph, find list of vertices adjacent to it:

verAdj=Transpose[Through[{VertexList,AdjacencyList}[g]]]

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

Return stabilizer list:

stabilizers=QuantumOperator[{"X"->#1,"Z"->#2}]&@@@verAdj

Apply above list of stabilizers and show that the result (graph state being transformed by a stabilizer) is still the same as the original state (that is why it is called stabilizer):

Thread[Through[stabilizers[ψf]]==ψf]