Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 10E / c / Solution

The amplitude at $y$ is proportional to $e^{2\pi i{x}_{0}y/8}{\sum }_{j=0}^{A-1}{e}^{2\pi ijry/8}$. The geometric sum vanishes unless $ry\equiv 0(\mathrm{mod}8)$, i.e. unless $y$ is a multiple of $A=8/r$. At those $r$ outcomes its modulus is $1/\sqrt{r}$, so each probability is $1/r$. The phase contains $x_0$, but measurement probabilities do not.

Solved by gpt-5.6-sol high.