The classical wavelet transform has been widely applied in the information processing field. It implies that quantum wavelet transform (QWT) may play an important role in quantum information processing. This chapter firstly describes the iteration equations of the general QWT using generalized tensor product. Then, Haar QWT (HQWT), Daubechies D4 QWT (DQWT), and their inverse transforms are proposed respectively. Meanwhile, the circuits of the two kinds of multi-level HQWT are designed. What's more, the multi-level DQWT based on the periodization extension is implemented. The complexity analysis shows that the proposed multi-level QWTs on 2n elements can be implemented by O(n3) basic operations. Simulation experiments demonstrate that the proposed QWTs are correct and effective.
TopSuppose that 
is a kernel matrix of the general wavelet, then, the (k+1)-level iteration of discrete wavelet transform is defined by (Ruch & Van Fleet, 2011),
(6.1) where the iteration matrix
is
(6.2) and
is a matrix with blocks
on the main diagonal and zeros elsewhere.
is a 2
m×2
m identity matrix
The iteration equations of 
and 
can be written as
,
(6.3) and
.
(6.4)From (5.12), we have
(6.5)Combining (6.4) with (6.5), we give the (k+1)-level iteration of general quantum wavelet transform
(6.6) or
(6.7) where the initial value is
.
Since
and
(see (5.18)), we design the quantum circuits of
shown in Figure 1. The detail circuits of perfect shuffle permutations
and
are seen in Figure 3.