Suppose I have $N \times N$ matrices $A,B,C,D$ and I form a $2 \times 2$ block matrix
$$ X = \begin{pmatrix} A & B \\ C & D \end{pmatrix}.$$
Furthermore, I have that $X$ is unitary. In my application, $N$ is a parameter that scales exponentially in the problem size, so instead of diagonalization of $X$, which would be computationally infeasible, I would like to block diagonalize $X$ using a unitary $U$ to obtain a following representation of $X$:
$$ X = U^\ast \begin{pmatrix} D_1 & 0 \\ 0 & D_2 \end{pmatrix} U.$$
My question is twofold. First: is this, even in principle, possible to do; even though $X$ is diagonalizable, it is not clear to me that such a block diagonalized form exists. Secondly, if it is possible, is there a way to accomplish this without having to access the matrix elements of $A,B,C,D$? Either an analytical procedure or some form of optimization over a space of unitaries that is dependent on $A,B,C,D$? Naturally, my first instinct was to try to compute $\det(X - \lambda I)$ and proceed from there, but this doesn't seem to be possible in general.