Tensor-product eigenphase amplification ($U^{\otimes n}|\psi {⟩}^{\otimes n}=e^{2\pi in\varphi }|\psi {⟩}^{\otimes n}$)

An eigenstate with phase $\varphi$ becomes an eigenstate with phase $n\varphi$ under the tensor product of $n$ copies of the unitary operator. A promise $0\le \varphi <1/n$ prevents modulo-one aliasing. For $n=2^s$ and dyadic $\varphi =x/2^m$, 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.