Suppose you have an invertible matrix. How do you convert it into a circuit?
Matrices have dimensions $2^n \times 2^n$, so a circuit representation is desirable.
For example, the matrix below is a simple classical circuit — which is why it has no imaginary entries — but I'm not sure if it could be converted into a quantum circuit with ease.
[ 1 0 0 0 0 0 0 0 ] [ 0 0 0 0 0 0 1 0 ] [ 0 0 1 0 0 0 0 0 ]M = [ 0 1 0 0 0 0 0 0 ] [ 0 0 0 1 0 0 0 0 ] [ 0 0 0 0 1 0 0 0 ] [ 0 0 0 0 0 1 0 0 ] [ 0 0 0 0 0 0 0 1 ]