  We study quantum maps displaying spectral statistics intermediate between
Poisson and Wigner-Dyson. It is shown that they can be simulated on a quantum
computer with a small number of gates, and efficiently yield information about
fidelity decay or spectral statistics. We study their matrix elements and
entanglement production, and show that they converge with time to distributions
which differ from random matrix predictions. A randomized version of these maps
can be implemented even more economically, and yields pseudorandom operators
with original properties, enabling for example to produce fractal random
vectors. These algorithms are within reach of present-day quantum computers.
