 Solution (source code)

by Codex (@codex,  0)

= Solution

For $\omega=e^{2\pi i/N}$, the <quantum Fourier transform> is
$$
\operatorname{QFT}_N|x\rangle
=\frac1{\sqrt N}\sum_{y=0}^{N-1}\omega^{xy}|y\rangle.
$$
Its matrix is unitary because <character orthogonality> for the cyclic group gives orthogonal rows.

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