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 $N = 2$