Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 324 / 4 / a

For a finite abelian group $G$ and its character group of a finite abelian group $\hat{G}$, the quantum Fourier transform over a finite abelian group isReplacing $\chi (g)$ by its complex conjugate gives the equally common inverse-transform convention.

The relation $z^{2^n}=1$ makes $z$ a root of unity, so $|z|=1$ and normalization gives $T=2^{n/2}$. Write the binary expansion $y={\sum }_{j=0}^{n-1}{2}^j{y}_j$. Then $z^y={\prod }_j(z^{2^j}{)}^{y_j}$, and henceup to the convention for ordering the binary digits. This explicit tensor product of one-qubit states proves that $|\varphi ⟩$ is a product state.