How to encode eigenvalues of matrix $A$ in solving $A\vec{x} = \vec{b}$ using the HHL Algorithm

Question

I am trying to implement multiple parallel subroutines of HHL algorithm, each working on a different set of matrix $A$ (when solving for $x$, in $A\vec{x} = \vec{b}$), to find the expectation values of $|x\rangle$. The matrix $A$ is randomly generated Hermitian matrix, and may/may not be sparse (checking what happens, is also something I'm wondering).

I have the following queries:

1. If I understand correctly, I need to encode eigenvalues of a $2 \times 2$ matrix $A$, say $\lambda_1$ and $\lambda_2$ in the quantum register. Is that right? How do I do that?
2. Is the quantum register initialized with the binary values of $\lambda_1$ and $\lambda_2$? Does that mean 2 runs of HHL? Is finding Eigenvalues of matrix $A$ supposed to be a classical subroutine?

Answer 1

Finding the eigenvalues of $A$ is an intermediate part of the HHL algorithm (although it will not output them). It is a quantum routine known as phase estimation, for which you need to be able to implement a controlled-unitary evolution where the unitary is determined by $U=e^{iAt}$ for some $t$. You do not need to find them by any classical routine. However, you do need some prior information: a bound on the range that the eigenvalues can occupy (this determines the $t$ in the previous relation), and a bound on how close to zero these eigenvalues might be (this determines how large a register you need for performing the phase estimation).