CNOT Gate using Measurement-Based Quantum Computing

Implement a widely used 2 qubit gate with an alternative approach to quantum computation

A CNOT gate can be built using 4 qubits. Measuring two of them in an appropriate basis and performing corrections achieves the desired transformation.

Install the Quantum Framework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"]

Load the paclet:

In[2]:=

<<Wolfram`QuantumFramework`

Represent the graph that determines the initial cluster state. Qubits 2 and 3 will be measured:

In[3]:=

g=HighlightGraphGraph{34,32,31},

,{2,3},ImageSizeSmall

Out[3]=

Initialize a random 2 qubit state:

In[4]:=

inputState=QuantumState["RandomPure"[2]]

Out[4]=

QuantumState

Pure state	Qudits: 2
Type: Vector	Dimension: 4



Construct the cluster state representing the graph with the exception that the first two qubits must have the initial random state:

In[5]:=

clusterCircuit=QuantumCircuitOperator[Join[{inputState,"H"{3,4}},"CZ"#&/@List@@@EdgeList[g]]]

Out[5]=

QuantumCircuitOperator



Simulate an outcome from the measurement of qubits 2 and 3 in the X basis ( the["SimulatedMeasurement"] property does this considering the probability distribution of the measurement):

In[6]:=

qm=(QuantumMeasurementOperator["X",{2,3}]@clusterCircuit[])["SimulatedMeasurement"]

Out[6]=



−



−

〉

Build the state from the QuditName returned by SimulatedMeasurement. Project the 4 qubits state with the bra of that state to get a 2 qubit state:

In[7]:=

projectedState=With[{s=QuantumState[QuantumBasis["X",2]["Association"][qm]//Flatten]},QuantumOperator[s["Dagger"],{2,3}]@clusterCircuit[]]

Out[7]=

QuantumState

Pure state	Qudits: 2
Type: Vector	Dimension: 4



Apply Pauli corrections to the state using the name of the measured state. If qubit 2 was

thestate|

〉

apply a Z correction, and if qubit 3 was the state

〉

correct with an X gate:

In[8]:=

correctedState=With[{res=qm["Name"]〚All,2,2〛},QuantumCircuitOperator[{If[res〚1〛-1,"Z"{1,2},"I"],If[res〚2〛-1,"X"2,"I"2]}]@projectedState]

Out[8]=

QuantumState

Pure state	Qudits: 2
Type: Vector	Dimension: 4



Check if the corrected state corresponds to the CNOT gate acting on the initial state:

In[9]:=

(cnotState=QuantumOperator["CNOT"]@inputState)correctedState

Out[9]=

True

Compare the state-vectors:

In[10]:=

globalPhaseExponent=BitAnd@@(Boole[#-1]&/@qm["Name"]〚All,2,2〛);

globalPhaseExponent

(-1)

2correctedState["StateVector"]-cnotState["StateVector"]//Normal//Chop

Out[10]=

{0,0,0,0}

In[168]:=

The states are the same up to a global phase that has no physical meaning. In this case, the projections change the normalization by a factor of 2, and an extra minus sign can appear due to the corrections. This verifies that the CNOT gate can be implemented in the measurement-based computing paradigm.

External Links

Wolfram/QuantumFramework

Source Metadata

Citation:

Publisher Information

Contributed by: Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

CNOT Gate using Measurement-Based Quantum Computing

External Links

Source Metadata

See Also

Publisher Information