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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?
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  • $\begingroup$ If you’re randomly generating A, why not randomly generate its inverse instead and save yourself the bother of running HHL? $\endgroup$ – DaftWullie Feb 6 '20 at 6:41
  • $\begingroup$ Thanks for your comment, DaftWullie. For now, I'm just testing on some random small size matrices, I intend to scale this to larger size matrices, and secondly I want to implement a quantum version inverse finding to learn doing it correctly. $\endgroup$ – viliyar Feb 6 '20 at 7:59
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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).

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