\displaystyle\ket{\varphi_j} = \frac{1}{\sqrt{N}} \sum_{i=0}^{N-1} \omega_N^{i \cdot j} \ket{i}$\omega$: [[roots-of-unity]]\displaystyle\mathbb{F}_N = \sum_{j} \ket{\varphi_j} \bra{j}$\mathbb{F}^\intercal = \mathbb{F} \Rightarrow \mathbb{F}^{-1} = \overline{\mathbb{F}}$N = 2$