Quantum computation and analysis of Wigner and Husimi functions: toward   a quantum image treatment
  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. 
M. Terraneo, B.Georgeot and D. L. Shepelyansky