= Tensor-product eigenphase amplification
{title2=$U^{\otimes n}|\psi\rangle^{\otimes n}=e^{2\pi in\phi}|\psi\rangle^{\otimes n}$}
An <eigenstate> with phase $\phi$ becomes an eigenstate with phase $n\phi$ under the tensor product of $n$ copies of the <unitary operator>. A promise $0\le\phi<1/n$ prevents modulo-one aliasing. For $n=2^s$ and dyadic $\phi=x/2^m$, <exact quantum phase estimation> then needs $m-s$ phase bits and $2^{m-s}-1$ queries to a supplied joint controlled oracle. This is a saving in joint-oracle calls, not in the total number of individual oracle invocations needed to build that joint operation.
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