I'm studying the HHL algorithm from the qiskit textbook, but I don't understand what $\lambda$ we have to use. If my matrix $A$ has eigenvalues that can be written in a binary representation, then, after QPE, in that register we just have an $n$-bit representation of the eigenvalues. Otherwise, in that register, we have an $n$-bit approximation, $2^n \frac{\lambda t}{2\pi}$. When I do controlled rotation, I have to calculate $2\arcsin(C/\lambda_j)$. My question is, which $\lambda_j$'s are the real eigenvalues of $A$: the values stored in the register or only $ \frac{\lambda t}{2\pi}$? My question arises because in this paper the authors use the exact eigenvalues, which can be written in a binary representation, but, the textbook uses $\frac{\lambda t}{2\pi}$, wherein this case, the eigenvalues cannot be written in a binary notation. Can't I always use the value inside the register $(2^n \frac{\lambda t}{2\pi})$ regardless of the representation?



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