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

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we get the following result:

enter image description here

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

enter image description here

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:

enter image description here

Then we can plug in:

enter image description here

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

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  • $\begingroup$ see quantumcomputing.stackexchange.com/questions/12390/… $\endgroup$ – Sam Palmer Mar 16 at 13:32
  • $\begingroup$ 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 $r$. $\endgroup$ – Sam Palmer Mar 16 at 13:46
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    $\begingroup$ 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? $\endgroup$ – mrW Mar 16 at 13:51
  • $\begingroup$ it happens to all of us! $\endgroup$ – Sam Palmer Mar 16 at 14:28
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The summation in the initial wave function should be $|\phi\rangle = \frac{1}{\sqrt{m}}\sum^{m-1}_0$

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