= Positive-phase fractional power of a unitary operator
{title2=$U^{1/M}_{+}=\sum_j e^{2\pi i\phi_j/M}|v_j\rangle\langle v_j|$}
Choose eigenphase representatives $0\leq\phi_j<1$ for a <unitary operator> $U$ and a positive integer $M$. Its positive-phase fractional power acts on each <eigenstate> by $e^{2\pi i\phi_j/M}$. This branch uses arguments in $[0,2\pi)$ and can differ from the usual complex principal branch using $(-\pi,\pi]$. For exactly dyadic phases, <exact quantum phase estimation> stores each label in a coherent phase register; bitwise <phase gates> multiply it by the required phase, and inverse phase estimation removes the labels. Keeping the phase register unmeasured preserves arbitrary <eigenstate> superpositions and resets the <ancilla qubits>. With only controlled-$U$ and controlled-$U^{-1}$ primitives, the direct method uses $2(2^n-1)$ oracle calls for $n$ exact phase bits.
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