# How to effectively compute eigenvalue rotation in HHL

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 ?

• Hello, did you check this tutorial from Qiskit directly? qiskit.org/textbook/ch-applications/hhl_tutorial.html
– Lena
Dec 9, 2020 at 17:50
• Yes, I read the all tutorial but they didn't explain how to compute eigenvalue rotation without lost of generality. In their example developed in part B, they apply a $2-$controlled rotation by $01$ and $10$ but it's a particular case. Dec 9, 2020 at 18:00

There is a new approach that will be merged soon in qiskit terra (here for the PR) that uses polynomial approximation to compute $$\arcsin(C/\lambda)$$, and asymptotically this would be the efficient implementation.
In practice if you are solving a $$2\times 2$$ matrix or a very small system it would be better to hard code the rotations.