Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 49 / 4 / b / Solution

The inverse quantum Fourier transform in the original PDF has matrix elements $N^{-1/2}{e}^{-2\pi iky/N}$ multiplying $|y⟩⟨k|$. The converted TeX incorrectly repeats $y$ in the bra; using the actual PDF operator,If $y=a$, every summand of the inner sum is one, so it equals $N$. If $y\ne a$, let $q=e^{2\pi i(a-y)/N}$. Then $q\ne 1$ and $q^N=1$, and the finite geometric series gives ${\sum }_{x=0}^{N-1}{q}^x=(1-q^N)/(1-q)=0$. This is orthogonality of roots of unity. HenceA quantum measurement in the computational basis consequently returns the integer $a$ with probability one.