Before reading on, let me acknowledge that this problem is solveable generally, however I am interested in knowing if a certain form of solution exists.
If I have a square complex unitary $n\times n$ matrix $A$, and for a given set of $n$ complex orthonormal column vectors $\{u_i\}$ of length $n$ ($i=1,...,n$), I know the set of also orthonormal column vectors $\{v_i\}$ where $v_i=Au_i$. I want to know if there is a formula to compute elements of $A$ knowing just $\{v_i\}$ and $\{u_i\}$.
I know generally this is solvable as this is a linear system, but it is not my interest to solve a system of $n$ linear equations. A numerical method would be sufficient, but algebraic best.
If $\mathrm{v}_i = \mathrm{A} \mathrm{u}_i$, then
$$\mathrm{V} := \begin{bmatrix} | & | & & | \\ \mathrm{v}_1 & \mathrm{v}_2 & \cdots & \mathrm{v}_n\\ | & | & & |\end{bmatrix} = \begin{bmatrix} | & | & & | \\ \mathrm{A} \mathrm{u}_1 & \mathrm{A} \mathrm{u}_2 & \cdots & \mathrm{A} \mathrm{u}_n\\ | & | & & |\end{bmatrix} = \mathrm{A} \begin{bmatrix} | & | & & | \\ \mathrm{u}_1 & \mathrm{u}_2 & \cdots & \mathrm{u}_n\\ | & | & & |\end{bmatrix} = \mathrm{A} \mathrm{U}$$
Hence, $\mathrm{V} \mathrm{U}^* = \mathrm{A} \underbrace{\mathrm{U} \mathrm{U}^*}_{= I_n}$ and, thus, $\mathrm{A} = \mathrm{V} \mathrm{U}^*$.