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Quantum phase estimation is usually explained via repeated applications of controlled $U^{2^k}$ gates in order to realize the map

$$ |j\rangle|\psi\rangle \mapsto |j\rangle U^j|\psi\rangle $$

If I assume that $U$ can be implemented as a quantum circuit, is there a general method to compute $U^{2^k}$ efficiently, or do I need to assume that $U^{2^k}$ can be implemented efficiently for all $k=0,\ldots,n$?

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The phase estimation algorithm assumes that you have an efficient way to calculate $U^{2^k}$. Generically, this may place a limit on $k$ and $2^k$, the precision to which you wish to calculate the phase.

This point is indeed often elided over. If you wish very (very) accurate calculation of the phase $\phi$ of your eigenstate $|\psi\rangle$ then you should be prepared to repeatedly run controlled versions of $U$ many times.

Nonetheless in the prototypical phase estimation algorithm used for factoring (in Shor's algorithm), we have, for very interesting number-theoretical reasons related to repeated squaring, a situation where we can "fast-forward" the calculation of $U^{2^k}$. Aharanov and Gilyen briefly touch on this point in a lecture at the Israeli Institute for Advanced Studies.

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