In this section, multi-level 1-dimensional quantum wavelet packet transforms (1D-QWPTs) are introduced. These 1D-QWPTs include 1-dimensional general quantum wavelet packet transform, 1-dimensional Haar quantum wavelet packet transform (1D-HQWPT), and 1-dimensional Daubechies D4 quantum wavelet packet transform (1D-D4QWPT).
Let 
be a wavelet kernel matrix. Then, the (k+1)-level iteration of a discrete wavelet packet transform is defined by (Ruch, & Van Fleet, 2011)
(7.1) where
with
j=1,…,
k is a matrix with 2
j blocks of
on the main diagonal and zeros elsewhere. We infer the following equations,
(7.2)According to the generalized tensor product in (5.12), we have
.
(7.3)The iteration of QWPT is given by
(7.4) with the initial value
. Its implemented circuit is shown in Figure 1 (a). The inverse of
is calculated by
(7.5) with the initial value
. The implemented circuit of
is shown in Figure 1 (b).
Figure 1. The implemented circuits of 
and 
.