Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 15D / b

The powers of $3$ modulo $10$ areThey then repeat, and none of the first three positive exponents returns $1$. Hence the least period is $\boxed{r=4}$.

Each of the four valuesoccurs for three of the twelve inputs. The corresponding basis states in the first register are orthogonal, so the Born rule gives

The value $3$ occurs at $x=1,5,9$. Projecting onto that measurement outcome and normalizing leaves the periodic coset state

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 $\omega =e^{2\pi i/12}$,Consequently,