Quantum Fourier Transform

Quantum Fourier Transform:

\[\sum_{j=0}^{N-1}{x_j|j\rangle}\Rightarrow \sum_{j=0}^{N-1}{x_j\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}{e^{2\pi i\frac{jk}{N}}|k\rangle}} \]

Quantum Inverse Fourier Transform:

\[\sum_{j=0}^{N-1}{x_j|j\rangle}\Rightarrow \sum_{j=0}^{N-1}{x_j\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}{e^{-2\pi i\frac{jk}{N}}|k\rangle}} \]