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Quantum Fourier Transform:

Performing linear transformation on the 鈥渜uantum bits鈥� is called 鈥渜uantum Fourier transform鈥� (QFT).

It is similar to the 鈥渄iscrete Fourier transform鈥� (DFT) where it works on the quantum state鈥檚 vector amplitude.

The 鈥渃lassical DFT鈥� works on the vector of $\left(a_0,K,a_{N-1}\right)$and map it to the vector$\left(b_0,K,b_{N-1}\right)$.

It is defined by the following formula,$b_I=\left(\frac{1}{\sqrt{M}}\underset{j=0}{\overset{M-1}{鈭�}}{a}_j{w}^{jI}\right)$

Here, the value of w is w= $e^{\frac{2蟺i}{M}}$and it is the $N^{th}$root of unity.

Similar to DFT, QFT works on the quantum state $\underset{j=0}{\overset{M-1}{鈭�}}{a}_j\overline{)j>}$and map it to the quantum state.

$\underset{j=0}{\overset{M-1}{鈭�}}{b}_j\overline{)j>.}$

It is defined by the following formula.

localid="1658904981124" $\overline{)j>}=\left(\frac{1}{\sqrt{M}}\underset{j=0}{\overset{M-1}{鈭�}}{w}^{jI}\overline{)J>}\right)$Therefore,

$QFT\left(j\right)=\left(\frac{1}{\sqrt{M}}\underset{j=0}{\overset{M-1}{鈭�}}{w}^{jI}\overline{)j>}\right)$