Quantum Wavelet Transforms

Quantum Wavelet Transforms

Copyright: © 2021 |Pages: 28
DOI: 10.4018/978-1-7998-3799-2.ch005
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Abstract

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.
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General Quantum Wavelet Transform

Suppose that 978-1-7998-3799-2.ch005.m01 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),

978-1-7998-3799-2.ch005.m02
(6.1) where the iteration matrix 978-1-7998-3799-2.ch005.m03 is
978-1-7998-3799-2.ch005.m04
(6.2) and
978-1-7998-3799-2.ch005.m05
is a matrix with blocks 978-1-7998-3799-2.ch005.m06 on the main diagonal and zeros elsewhere. 978-1-7998-3799-2.ch005.m07 is a 2m×2m identity matrix

The iteration equations of 978-1-7998-3799-2.ch005.m08 and 978-1-7998-3799-2.ch005.m09 can be written as

978-1-7998-3799-2.ch005.m10
,(6.3) and

978-1-7998-3799-2.ch005.m11
.(6.4)

From (5.12), we have

978-1-7998-3799-2.ch005.m12
(6.5)

Combining (6.4) with (6.5), we give the (k+1)-level iteration of general quantum wavelet transform

978-1-7998-3799-2.ch005.m13
(6.6) or
978-1-7998-3799-2.ch005.m14
(6.7) where the initial value is 978-1-7998-3799-2.ch005.m15.

Since

978-1-7998-3799-2.ch005.m16
and
978-1-7998-3799-2.ch005.m17
(see (5.18)), we design the quantum circuits of 978-1-7998-3799-2.ch005.m18 shown in Figure 1. The detail circuits of perfect shuffle permutations 978-1-7998-3799-2.ch005.m19 and 978-1-7998-3799-2.ch005.m20 are seen in Figure 3.

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