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I am interested in HHL algorithm and despite all the problems related to current technologies, I am trying to understand how it has to be implemented.

I have seen from these two available circuits (arXiv:1804.03719, arXiv:1811.01726) that when you define a circuit it is necessary to apply a certain rotation of the ancilla qubit based on the eigenvalues. For what I understood from the original paper, the algorithm "extracts" the information about eigenvalues using the quantum phase estimation algorithm but this part is missing in the circuits.

My doubt is if I have to decompose my matrix to find eigenvalues I lose the possible advantage derived from the algorithm; is that correct?

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