Quantum Wavelet Packet Transforms

Quantum Wavelet Packet Transforms

Copyright: © 2021 |Pages: 25
DOI: 10.4018/978-1-7998-3799-2.ch006

Abstract

Quantum wavelet packet transform (QWPT) may play an important role in quantum information processing. In this chapter, the authors design quantum circuits of a generalized tensor product (GTP) and a perfect shuffle permutation (PSP). Next, they propose multi-level and multi-dimensional (1D, 2D and 3D) QWPTs, including Haar QWPT (HQWPT), 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, they analyze the quantum costs and the time complexities of the proposed QWPTs and obtain precise results. The time complexities of HQWPTs is at most six basic operations on 2n elements, which illustrates high efficiency of the proposed QWPTs.
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1-Dimensional Quantum Wavelet Packet Transforms

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).

1-Dimensional General Quantum Wavelet Packet Transform

Let 978-1-7998-3799-2.ch006.m01 be a wavelet kernel matrix. Then, the (k+1)-level iteration of a discrete wavelet packet transform is defined by (Ruch, & Van Fleet, 2011)

978-1-7998-3799-2.ch006.m02
(7.1) where 978-1-7998-3799-2.ch006.m03 with j=1,…,k is a matrix with 2j blocks of 978-1-7998-3799-2.ch006.m04on the main diagonal and zeros elsewhere. We infer the following equations,

978-1-7998-3799-2.ch006.m05
(7.2)

According to the generalized tensor product in (5.12), we have

978-1-7998-3799-2.ch006.m06
.(7.3)

The iteration of QWPT is given by

978-1-7998-3799-2.ch006.m07
(7.4) with the initial value 978-1-7998-3799-2.ch006.m08. Its implemented circuit is shown in Figure 1 (a). The inverse of 978-1-7998-3799-2.ch006.m09is calculated by
978-1-7998-3799-2.ch006.m10
(7.5) with the initial value 978-1-7998-3799-2.ch006.m11. The implemented circuit of 978-1-7998-3799-2.ch006.m12 is shown in Figure 1 (b).

Figure 1.

The implemented circuits of 978-1-7998-3799-2.ch006.m13 and 978-1-7998-3799-2.ch006.m14.

978-1-7998-3799-2.ch006.f01

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