Quantum Fourier Transform in the Period Finding Problem

I am trying to prove that when applying the inverse QFT to the following state:

we get the following result:

However, I get a wrong prefactor. Can anyone tell me where I went wrong? Here my calculations:

Here, the last equality comes by realizing that y is an integer and thus all terms of the second bracket give 1. The first bracket can be evaluated as follows:

Then we can plug in:

Obviously I should get 1/sqrt(r) not sqrt(r), but I don't find my mistake...

• Mar 16, 2021 at 13:32
• your issue is that in the initial wave function the sum should be to to $m-1$ not $rm - 1$. Then the rest of your proof is correct as you won't end up with the additional factor $r$. Mar 16, 2021 at 13:46
• ahhh sometimes you do not see the forest for the trees... Thanks a lot! Should I delete this post since it is such a stupid mistake?
– mrW
Mar 16, 2021 at 13:51
• it happens to all of us! Mar 16, 2021 at 14:28

The summation in the initial wave function should be $$|\phi\rangle = \frac{1}{\sqrt{m}}\sum^{m-1}_0$$