Solution
= Solution
Here $N=12$, $r=4$, $A=N/r=3$, and $x_0=1$. The supplied <quantum Fourier transform of a periodic coset state> gives, with $\omega=e^{2\pi i/12}$,
$$
\begin{aligned}
\operatorname{QFT}_{12}|\alpha\rangle
&=\frac12\sum_{l=0}^3\omega^{3l}|3l\rangle\\
&=\frac12\left(|0\rangle+i|3\rangle-|6\rangle-i|9\rangle\right).
\end{aligned}
$$
Consequently,
$$
\boxed{
\Pr(c)=
\begin{cases}
\frac14,&c\in\{0,3,6,9\},\\
0,&\text{otherwise}.
\end{cases}}
$$