Graph-state circuit

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

Examples

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:

In[1]:=

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

Out[1]=

Create the corresponding quantum circuit:

In[2]:=

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

Out[2]=

QuantumCircuitOperator



Circuit diagram:

In[3]:=

qc["Diagram"]

Out[3]=

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

In[4]:=

ψf=qc[]

Out[4]=

QuantumState

Pure state	Qudits: 4
Type: Vector	Dimension: 16
Picture: Schrödinger



In[5]:=

ψfQuantumState[{"Graph",g}]

Out[5]=

True

Show qubits connected via an edge are entangled:

In[6]:=

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

Out[6]=

{True,True,True,True,True}

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

In[7]:=

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

Out[7]=

{{1,{2,3}},{2,{1,3,4}},{3,{1,2,4}},{4,{2,3}}}

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:

In[8]:=

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

Out[8]=

QuantumOperator

Picture: Schrödinger	Arity: 3
Dimension: 8→8	Qudits: 3→3

,QuantumOperator

Picture: Schrödinger	Arity: 4
Dimension: 16→16	Qudits: 4→4

,QuantumOperator

Picture: Schrödinger	Arity: 4
Dimension: 16→16	Qudits: 4→4

,QuantumOperator

Picture: Schrödinger	Arity: 3
Dimension: 8→8	Qudits: 3→3



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):

In[9]:=

Thread[Through[stabilizers[ψf]]ψf]

Out[9]=

{True,True,True,True}

External Links

QuantumFramework | Wolfram Language Paclet Repository

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Contributed By: Mads Bahrami

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Graph-state circuit

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