Quantum Haar Wavelet Transforms and Their Applications

Fourier transform has been shown to be a powerful tool in many area of science. However, there is another class of unitary transforms, the wavelet transforms, which are as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and very useful for image processing and data compressiyon. In this paper, we construct quantum algorithms for Haar wavelet transforms and show its application in analyzing the multi-scale structure of the dynamical system by the Logistic Map (x! x(1 x)).