Graph-state circuit

Construct a graph-state (also called as a cluster state) circuit corresponding to a specified graph

Example ContentResourceCreateNotebook

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]

Supporting Data and Definitions

(Your data can be Dataset[…], EntityStore[…], Image[…], Audio[…] or any other expression)

(xxxx can be your data, or File[…], CloudObject[…] or LocalObject[…] that contains it.)

Thumbnail Image

Source & Additional Information

Contributed By

Mads Bahrami

KeywordsResourceCreateNotebook

Quantum circuit

Quantum state

Graph state

Cluster state

CategoriesResourceCreateNotebook

Related Documentation Pages

Related Resource ObjectsResourceCreateNotebook

Resource Name (resources from any Wolfram repository)

Original Source References and AttributionsResourceCreateNotebook

Source, reference or citation

LinksResourceCreateNotebook

QuantumFramework | Wolfram Language Paclet Repository

CompatibilityResourceCreateNotebook

Wolfram Language VersionResourceCreateNotebook

13.2+

Operating SystemResourceCreateNotebook

EnvironmentsResourceCreateNotebook

Cloud SupportResourceCreateNotebook

Required FeaturesResourceCreateNotebook

Author NotesResourceCreateNotebook

Additional information about limitations, issues, etc.