# HHL phase estimation step

I have got an HHL circuit that looks as follows:

In the phase estimation part we are trying to find the eigenvalues of the matrix A. But what is the role of the piece of circuit hightlighted below?

As I understant, what are we doing here is to try apply a gate that affects the same as a matrix A multiplied on some angle $$\pi / 2^n$$. And due to the phase kickback, the q1-q3 qubits stores some phases. What these phases stand for? I read somewhere that this is "phase representation of the eigenvalues" - but why is it so? And why do we need this anlge $$\pi / 2^n$$?

• This paper may help you: arxiv.org/abs/2108.09004 Mar 9, 2023 at 17:33
• It also does not explain what this step mean. It describes what to do but not why Mar 10, 2023 at 6:12
• This step should be the phase estimation step. Typically, you would see controlled-U's to exponentiated matrices $U$ such as $U^{2^0}, U^{2^1}, U^{2^2}$, with each $U = e^{i A t}$. In the description, the $t$ is then chosen such that the eigenvalues map to full integers. So I don't know why there would be those fractions of $\pi$ in your circuit. Does it work? Here is my implementation of that paper. Mar 10, 2023 at 6:34
• But why do we apply this unitary? Why do we exponentiate the matrix and apply it's control version? What is the logic behind? What is the result before Fourier step and after applying U in power of 2? Why do not we do anything else but this? Mar 10, 2023 at 6:41
• There are plenty of articles explaining phase estimation, see for example here. Mar 10, 2023 at 14:44