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I'm going to change the notation slightly to make it a bit easier for me: I'll assume it's an arbitrary group of order $N$. There is a minor mistake in your first equation, which is that the QFT will have different moduli for the different integers. That is, $a$ is an $n$-bit register, so the QFT will produce a phase modulo $2^n$, whereas $b$ is a $\lceil \...

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