For a <finite abelian group> $G$ and its <character group of a finite abelian group> $\widehat G$, the <quantum Fourier transform over a finite abelian group> is
$$
\operatorname{QFT}_G|g\rangle
=\frac1{\sqrt{|G|}}\sum_{\chi\in\widehat G}\chi(g)|\chi\rangle.
$$
Replacing $\chi(g)$ by its complex conjugate gives the equally common inverse-transform convention.

Solution

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

{title2=$\operatorname{QFT}_G$}

For a <finite abelian group> $G$ with character group $\widehat G$, the quantum Fourier transform is
$$
|g\rangle\longmapsto\frac1{\sqrt{|G|}}\sum_{\chi\in\widehat G}\chi(g)|\chi\rangle,
$$
up to complex conjugation according to the transform convention.

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 Incoming links: Quantum Fourier transform over a finite abelian group