# How can I express controlled unitary operation in QPE of this implementation of HHL?

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?

• Hi glS, sorry there were a few typos, I edited the question. Jun 6 '19 at 7:01