I have found this implementation of HHL, and I don't understand why the controlled unitary operation is expressed in the form of $\exp(i t_0 A/2)$ and $\exp(i t_0 A/4)$.
The rotation of $\pi$ and $\pi/2$ are derived by the use of $\exp(i t_0 A/2)$ and $\exp(i t_0 A/4)$, in particular using the diagonal matrix which contains the eigenvalues. But usually, the controlled operations of QPE step is done using the matrices $A^{2^0}, A^{2^1}, … , A^{2^k}$. Where am I wrong?
I have also another question about the representation of the unitary operation: we can we express the unitary operation using only the matrix of the eigenvalues because $b$ is expressed on the same (standard) basis of the eigenvectors, is that correct?
