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For an HHL algorithm implemented exactly as depicted in Figure 2 of this paper by Dutta et. al (https://arxiv.org/abs/1811.01726), how do I go about changing the eigenvalues that they use? Obviously, if the eigenvalues are changed, then the matrix A should be changed appropriately, but this is more of a question about how the controlled y rotations change.

In this paper, if the eigenvalues are set equal to $\lambda = [1\ 2\ 4\ 8]$, then the controlled y rotations are implemented from left to right with the y rotations being set equal to $R_y(\frac{8\pi}{\lambda_1 2^{r}})=R_y(\frac{8\pi}{2^{r}})$, $R_y(\frac{8\pi}{\lambda_2 2^{r}})=R_y(\frac{4\pi}{2^{r}})$, $R_y(\frac{8\pi}{\lambda_3 2^{r}})=R_y(\frac{2\pi}{2^{r}})$, and $R_y(\frac{8\pi}{\lambda_4 2^{r}})=R_y(\frac{\pi}{2^{r}})$, respectively. Here, $r$ is an arbitrary integer whose value effects the accuracy of the final outcome. But now let's say that I want to change the eigenvalues to $\lambda = [1\ 3\ 4\ 8]$, would the y rotation corresponding to the second eigenvalue as well as the state defined as $|j_2\rangle$, be given by $R_y(\frac{8\pi}{\lambda_2 2^{r}})=R_y(\frac{8\pi}{3\times 2^{r}})$?

If this is not the case, what is the appropriate way to change this setup to get this calculation to work? Are there any swap gates that I need to add? Do I need to implement some sort of method to run this algorithm while increasing the number of clock qubits and if so, how do I do this?

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