Complete quantum circuit of Haar wavelet based MRA

Wavelet analysis has applications in many areas, such as signal analysis and image processing. We propose a method for generating the complete circuit of Haar wavelet based MRA by factoring butterfly matrices and conditional perfect shuffle permutation matrices. The factorization of butterfly matrices is the essential part of the design. As a result, it is the key point to obtain the circuits of I_2t ⊕W ⊕I_2n-2t-2. In this paper, we use a simple means to develop quantum circuits for this kind of matrices. Similarly, the conditional permutation matrix is implemented entirely, combined with the scheme of Fijany and Williams. The circuits and the ideas adopted in the design are simple and intelligible.