In this paper, there is a diagram explaining how HHL works, which I attached below:
My question is, for the rotation part, how do you write $R$ as a combination of universal set of gates without knowing the eigenvalues a priori? Or what is a generic way of writing $R$ without doing any classical pre-computation regarding the eigenvalues?
In most of the implementations that I found online, it is assumed that the eigenvalues are known, and then one can find the angles of rotation by calculating for each eigenvalue: $\theta_i = arccos(\frac{C}{\lambda_i})$. After that, one can append for the ancilla register multiple $R_y$ gates with the aforementioned angles of rotation as arguments.
Thanks in advance for the help!