The multi-level and multi-dimensional quantum wavelet packet transforms

The classical wavelet packet transform has been widely applied in the information processing field. It implies that the quantum wavelet packet transform (QWPT) can play an important role in quantum information processing. In this paper, we design quantum circuits of a generalized tensor product (GTP) and a perfect shuffle permutation (PSP). Next, we propose multi-level and multi-dimensional (1D, 2D and 3D) QWPTs, including a Haar QWPT (HQWPT), a D4 QWPT (DQWPT) based on the periodization extension and their inverse transforms for the first time, and prove the correctness based on the GTP and PSP. Furthermore, we analyze the quantum costs and the time complexities of our proposed QWPTs and obtain precise results. The time complexities of HQWPTs is at most 6 on 2n elements, which illustrates high-efficiency of the proposed QWPTs. Simulation experiments demonstrate that the proposed QWPTs are correct and effective.