# 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 $`N = 2`$