Fourier Vector

• quantum-computing
• `\displaystyle\ket{\varphi_j} = \frac{1}{\sqrt{N}} \sum_{i=0}^{N-1} \omega_N^{i \cdot j} \ket{i}`
  • `\omega`: roots-of-unity
• quantum-fourier-transform: `\displaystyle\mathbb{F}_N = \sum_{j} \ket{\varphi_j} \bra{j}`
  • `\mathbb{F}^\intercal = \mathbb{F} \Rightarrow \mathbb{F}^{-1} = \overline{\mathbb{F}}`
• fourier-basis: special case when `N = 2`