Why do we need deterministic measurement matrices in compressed sensing?
As far as I know there are two reasons:

In sensing part: For practical implementation, usage of random matrices
is hard, so people try to come up with simpler matrices that are fixed,
this is thought to be helpful for designing efficient hardware.
In reconstruction part: To find deterministic matrices which their
structure can be exploited for faster signal recovery. Using a
structured matrix, might pave the way for efficient recovery. The
goal is to achieving something like FFT for compressive recovery.