I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ with exactly $M$ elements.

Let $U(s,s')$ be the $M\times M$ submatrix of U whose rows are specified by the elements of $s\in\mathcal S$ and whose columns are specified by the elements of $s'\in\mathcal S$.

My data is a set of $I(s,s')\in\mathbb Q$ such that $$ \det U(s,s') = e^{i \pi I(s,s')}.$$

Can I find the full matrix $U$? I would be interested also in understand why this problem might not have a solution. However, if the solution exists (for certain values of $M,N$), than I would like to find the algorithm to obtain $U$.