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I am wondering if the second Quantum Fourier Transform (QFT) in Shor's algorithm is necessary. I am probably missing a point but it seems that an offset elimination function would suffice to determine the period?

QFT eliminates the offset value and it changes the period from r to M/r, where M is a sufficiently large value >> r^2. What we do normally is to take a measurement and get a multiple of M/r. Doing this a couple of times we take the GCD of those measured values and obtain M/r, and then it is easy to obtain r from it.

But is changing the period necessary if there is no offset? Suppose that the period we are looking for is, say 5. Given a superposition with non-zero values only on the multiples of 5, we have a periodic sequence with period 5. Make a measurement and obtain a random multiple of 5. Do this constant number of times. Since all measured values will be multiples of 5, can't I determine r from it by finding their common divisor? So why do we need QFT? Isn't just an offset elimination function sufficient to find the period?

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Yes, if you could consistently eliminate the offset then that would be enough. But how are you going to do that?

The state's offset is the discrete logarithm of the value in the ancilla register. But you don't know how to compute the discrete logarithm efficiently classically. And the offset changes from sample to sample. And there are so many possible offsets (assuming you picked a big number to factor) that you won't see the same one twice.

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