Quantum Fourier sampling of powers of three modulo ten

The function $f(x)=3^x\mathrm{mod}10$ on ${\mathbb{Z}}_{12}$ has period four. Measuring function value $3$ in its uniform input-output state leaves $(|1⟩+|5⟩+|9⟩)/\sqrt{3}$. The quantum Fourier transform modulo twelve then returns each of $0,3,6,9$ with probability $1/4$.