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Why do the unitary gates on the measurement qubits have $2^n$? Why do we need to apply the unitary gates for any power at all? What would happen if we applied the controlled-$U$ only once, for example, for each control qubit, prior to the inverse QFT?

For example, from Wikipedia here.

Specifically the circuit diagram illustrated in Wikipedia.

Another resource that shows this is the Qiskit documentation.

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    $\begingroup$ Is your question why $2^n$ as opposed to some other power? The QFT factors nicely to determine a binary representation of the phase. Remember we also like to pad the (classical) FFT to a power of $2$. Cooley and Tukey (rediscovering the work of Gauss) . $\endgroup$
    – Mark S
    Feb 6 at 1:07
  • $\begingroup$ Why a power at all? Why not just U with no power for all unitary gates? $\endgroup$
    – mikanim
    Feb 6 at 8:18
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    $\begingroup$ @mikanim if you apply $U$ just once you will only learn one bit of the phase... not so useful. $\endgroup$
    – Condo
    Feb 8 at 18:02
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I also struggled with this question initially and I understood it really well after looking through these set of slides starting at "Quantum Phase Estimation" (page 120 in the pdf): https://drive.google.com/file/d/14G_0TwdxBFpI_Ylj5lb_imVtcnunrQcB/view?usp=sharing

But the short answer is that for every extra level of accuracy in binary, you need an extra power of 2 of unitaries.

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