Quantum Fourier Transform - Example

Example

Consider the quantum Fourier transform on 3 qubits. It is the following transformation:

,

where is a primitive eighth root of unity satisfying (since ).

The matrix representing this transformation on 3 qubits is

F_{2^3} = \frac{1}{\sqrt{2^3}} \begin{bmatrix} 1&1&1&1&1&1&1&1 \\ 1&\omega&\omega^2&\omega^3&\omega^4&\omega^5&\omega^6&\omega^7 \\ 1&\omega^2&\omega^4&\omega^6&\omega^8&\omega^{10}&\omega^{12}&\omega^{14} \\ 1&\omega^3&\omega^6&\omega^9&\omega^{12}&\omega^{15}&\omega^{18}&\omega^{21} \\ 1&\omega^4&\omega^8&\omega^{12}&\omega^{16}&\omega^{20}&\omega^{24}&\omega^{28} \\ 1&\omega^5&\omega^{10}&\omega^{15}&\omega^{20}&\omega^{25}&\omega^{30}&\omega^{35} \\ 1&\omega^6&\omega^{12}&\omega^{18}&\omega^{24}&\omega^{30}&\omega^{36}&\omega^{42} \\ 1&\omega^7&\omega^{14}&\omega^{21}&\omega^{28}&\omega^{35}&\omega^{42}&\omega^{49} \\ \end{bmatrix} = \frac{1}{\sqrt{2^3}} \begin{bmatrix} 1&1&1&1&1&1&1&1 \\ 1&\omega&\omega^2&\omega^3&\omega^4&\omega^5&\omega^6&\omega^7 \\ 1&\omega^2&\omega^4&\omega^6&1&\omega^2&\omega^4&\omega^6 \\ 1&\omega^3&\omega^6&\omega&\omega^4&\omega^7&\omega^2&\omega^5 \\ 1&\omega^4&1&\omega^4&1&\omega^4&1&\omega^4 \\ 1&\omega^5&\omega^2&\omega^7&\omega^4&\omega&\omega^6&\omega^3 \\ 1&\omega^6&\omega^4&\omega^2&1&\omega^6&\omega^4&\omega^2 \\ 1&\omega^7&\omega^6&\omega^5&\omega^4&\omega^3&\omega^2&\omega \\ \end{bmatrix} .

The 3-qubit quantum Fourier transform is the following operation:

This quantum circuit implements the quantum Fourier transform on the quantum state .

The quantum gates used in the circuit above are the Hadamard gate and the controlled phase gate .

As calculated above, the number of gates used is which is equal to 6, for n = 3.

Read more about this topic:  Quantum Fourier Transform

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