Positive-phase fractional power of a unitary operator ($U_{+}^{1/M}={\sum }_j{e}^{2\pi i{\varphi }_j/M}|v_j⟩⟨v_j|$)

Choose eigenphase representatives $0\le {\varphi }_j<1$ for a unitary operator $U$ and a positive integer $M$. Its positive-phase fractional power acts on each eigenstate by $e^{2\pi i{\varphi }_j/M}$. 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.