Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 10E (Quantum Information and Computation)

The powers areThus the least positive period is $\boxed{r=3}$.

The possible values are $y\in \{1,4,7\}$. Each occurs for four of the twelve equally weighted values of $x$, so

The equality $f(x)=4$ holds exactly for $x\equiv 1(\mathrm{mod}3)$. Projecting onto the measured second-register value and normalizing gives

With $\omega =e^{2\pi i/12}$, the quantum Fourier transform acts by(The opposite phase convention gives the same measurement probabilities.)

Applying the transform to the coset state gives amplitudefor $|c⟩$. The geometric sum vanishes unless ${\omega }^{3c}=1$, which means $c\equiv 0(\mathrm{mod}4)$. For $c=0,4,8$ its magnitude is $4/(2\sqrt{12})=1/\sqrt{3}$. Thus the quantum Fourier transform of a periodic coset state gives