Quantum Fourier transform toolbox
This research develops efficient quantum circuits for Fourier Transforms over non-abelian finite groups using advanced group theory like Mackey and Clifford theory. It achieves polynomial circuit sizes, such as $ ext{poly}(\log q)$ for $\text{GL}_2(\mathbb{F}_q)$, significantly improving complexity beyond previous methods. This provides practical, scalable quantum algorithms for complex group structures.