In the HHL algorithm, how do you efficiently do the $\lambda-$controlled rotation on the ancillary qubit ? It seems to me after reading around some answers that this can be done in two steps :
- First, we map $|\lambda\rangle\mapsto |\frac{1}{\pi}\arcsin(\frac{C}{\lambda})\rangle$, defining $|\frac{1}{\pi}\arcsin(\frac{C}{\lambda})\rangle$ to be a binary representation $|\frac{1}{\pi}\arcsin(\frac{C}{\lambda})\rangle$ with $m$ qubits.
- Then perform a controlled rotation $U_y(|\theta\rangle \otimes |0\rangle)\mapsto |\theta\rangle \otimes \big(\cos(\theta)|0\rangle + \sin{(\theta})|1\rangle\big)$ where $U_y$ is simply $$ U_y(|\theta\rangle \otimes |0\rangle) = \prod_{j=1}^m (I^{\otimes^m}\otimes R_y(2\pi\theta_j/2^j)) $$ i.e. a sequence of controlled rotations where we successively halve the angle of rotation conditionally of the digits of the binary representation of $\theta$.
My question is the following how can one implement efficiently the first step in an environment such as Qiskit ?
