  We study the efficiency of quantum algorithms which aim at obtaining phase
space distribution functions of quantum systems. Wigner and Husimi functions
are considered. Different quantum algorithms are envisioned to build these
functions, and compared with the classical computation. Different procedures to
extract more efficiently information from the final wave function of these
algorithms are studied, including coarse-grained measurements, amplitude
amplification and measure of wavelet-transformed wave function. The algorithms
are analyzed and numerically tested on a complex quantum system showing
different behavior depending on parameters, namely the kicked rotator. The
results for the Wigner function show in particular that the use of the quantum
wavelet transform gives a polynomial gain over classical computation. For the
Husimi distribution, the gain is much larger than for the Wigner function, and
is bigger with the help of amplitude amplification and wavelet transforms. We
also apply the same set of techniques to the analysis of real images. The
results show that the use of the quantum wavelet transform allows to lower
dramatically the number of measurements needed, but at the cost of a large loss
of information.
