 Solution

Suppose the angle register stores a fixed-point expansion ${\theta }_i={\sum }_{r=1}^q{\theta }_{ir}{2}^{-r}{\theta }_{\max }$. Append a target qubit in $|0⟩$. For each angle bit $het{a}_{ir}$, apply to the target a controlled $R_y(2^{1-r}{\theta }_{\max })$. Rotations about the same axis commute, so their product is $R_y(2{\theta }_i)$ andWith $q=O(\mathrm{poly}\log N)$ retained bits, each controlled rotation decomposes into one- and two-qubit gates and the complete quantum variable rotation has polylogarithmic size. Thus the required branchwise map is implemented coherently for every $i$.