Output | vector $(y_0,y_1,\ldots,y_N-1)\in\mathbb{C}^N$ according to $y_k={\frac{1}{\sqrt{N}}}\sum_{n=0}^{N-1}x_{n}\omega_{N}^{-kn},$ $k=0,1,2,\ldots ,N-1$ | quantum state $\sum_{i=0}^{N-1}y_{i}|i\rangle$ according to $y_{k}={\frac{1}{\sqrt{N}}}\sum_{n=0}^{N-1}x_{n}\omega_{N}^{nk},$ $k=0,1,2,\ldots ,N-1$
|