Quantum Fourier Transform

The quantum Fourier transform (QFT) is a quantum analog of the classical discrete Fourier transform (DFT) and is a key component in many quantum algorithms, including Shor's algorithm. QFT circuits transform a quantum state from the computational basis to the Fourier basis.

Install and load the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"]Needs["Wolfram`QuantumFramework`"]

Out[1]=

PacletObject

Name: Wolfram/QuantumFramework

Version: 1.0.35



Use

QuantumCircuitOperator

to create the corresponding quantum Fourier transform circuit of 3-qubits:

In[2]:=

qft=QuantumCircuitOperator[{"Fourier",3}]

Out[2]=

QuantumCircuitOperator



Circuit diagram:

In[3]:=

qft["Diagram"]

Out[3]=

The matrix representation of the Fourier transform on three qubits:

In[4]:=

qft["Matrix"]//MatrixForm

Out[4]//MatrixForm=

1 2 2	1 2 2	1 2 2	1 2 2	1 2 2	1 2 2	1 2 2	1 2 2
1 2 2	π 4  2 2	 2 2	 π 4  2 2	- 1 2 2	- π 4  2 2	-  2 2	-  π 4  2 2
1 2 2	 2 2	- 1 2 2	-  2 2	1 2 2	 2 2	- 1 2 2	-  2 2
1 2 2	 π 4  2 2	-  2 2	π 4  2 2	- 1 2 2	-  π 4  2 2	 2 2	- π 4  2 2
1 2 2	- 1 2 2	1 2 2	- 1 2 2	1 2 2	- 1 2 2	1 2 2	- 1 2 2
1 2 2	- π 4  2 2	 2 2	-  π 4  2 2	- 1 2 2	π 4  2 2	-  2 2	 π 4  2 2
1 2 2	-  2 2	- 1 2 2	 2 2	1 2 2	-  2 2	- 1 2 2	 2 2
1 2 2	-  π 4  2 2	-  2 2	- π 4  2 2	- 1 2 2	 π 4  2 2	 2 2	π 4  2 2

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