This is a follow-up question to solve for parameters of a controlled $U_3$ gate. Suppose I have the unit vector $\hat{n} = (n_x,n_y,n_z)$ and an angle $\xi$ (assume we know both of them). Then, we could calculate $$ \begin{align} \begin{aligned} R_{\hat{n}}(\xi) &= \cos(\frac{\xi}{2})I-i\sin(\frac{\xi}{2})(\hat{n}\cdot\vec\sigma)\\ &= \cos(\frac{\xi}{2})I-i\sin(\frac{\xi}{2})\begin{bmatrix} n_z & n_x-in_y\\ n_x+in_y & -n_z \end{bmatrix}\\ &=\begin{bmatrix} \cos(\frac{\xi}{2})-i\sin(\frac{\xi}{2})n_z & -i\sin(\frac{\xi}{2})(n_x-in_y)\\ -i\sin(\frac{\xi}{2})(n_x+in_y) & \cos(\frac{\xi}{2})+i\sin(\frac{\xi}{2})n_z \end{bmatrix} \end{aligned} \end{align} $$ Then, an arbitrary single qubit rotation could be written as $U = e^{i\alpha}R_{\hat{n}}(\xi)$. Assume we also know the value of $\alpha$, how can we implement the controlled-$U$ in qiskit? I'm really confused with how can I utilize the (controlled) $U_3$ gate, which is defined as \begin{align} U_3(\theta,\phi,\lambda)= \begin{bmatrix} \cos\left(\frac{\theta}{2}\right) & -e^{i\lambda} \sin\left(\frac{\theta}{2}\right) \\ e^{i\phi} \sin\left(\frac{\theta}{2}\right) & e^{i(\lambda + \phi)} \cos\left(\frac{\theta}{2}\right) \end{bmatrix}. \end{align}
More specifically, if we want to implement the controlled version of \begin{align} U = e^{i\alpha}R_{\hat{n}}(\xi) \end{align} Which should be treated as $U_3$? $U$ or $R_{\hat n}(\xi)$? Any suggestions for figuring out the arguments of $U_3$? Do I need to worry about phase kickback?
Thanks a lot for the help!
