A complex $n \times n$ unitary operator has $n^2$ free real parameters. For example, a $2 \times 2$ unitary matrix can be parametrized as \begin{equation} \begin{pmatrix} e^{i(\alpha - \beta/2 - \delta/2)} \cos \frac{\gamma}{2} & -e^{i(\alpha - \beta/2 + \delta/2)} \sin \frac{\gamma}{2}\\ e^{i(\alpha + \beta/2 - \delta/2)} \sin \frac{\gamma}{2} & e^{i(\alpha + \beta/2 + \delta/2)} \cos \frac{\gamma}{2} \end{pmatrix}. \end{equation}
However, if a initial state is given as \begin{equation} |\psi_i \rangle= \begin{pmatrix} \cos \frac{\theta_i}{2} \\ \sin \frac{\theta_i}{2} e^{i\phi_i} \end{pmatrix}, \end{equation} and the desired output is \begin{equation} |\psi_f \rangle= \begin{pmatrix} \cos \frac{\theta_f}{2} \\ \sin \frac{\theta_f}{2} e^{i\phi_f} \end{pmatrix}. \end{equation} For both input and output, the global phase is not of interest and therefore omitted.
I would like to find a unitary operator $U$ that converts $|\psi_i \rangle$ to $|\psi_f \rangle$. Since the initial state is known, there are only $2$ parameters underdetermined. This indicates that the $U$ such that $|\psi_i \rangle \xrightarrow{U} |\psi_f \rangle$ contains only two independent parameters.
It seems that two parameters in the general $2\times 2$ unitary matrix are redundant.