Three-dimensional quantum wavelet transforms

doi:10.1007/s11704-022-1639-y

Front. Comput. Sci.

2023, Vol. 17

Issue (5) : 175905 https://doi.org/10.1007/s11704-022-1639-y

RESEARCH ARTICLE

Three-dimensional quantum wavelet transforms

Haisheng LI(

), Guiqiong LI(

), Haiying XIA(

)

College of Electronic and Information Engineering, Guangxi Normal University, Guilin 541004, China

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Abstract

Wavelet transform is being widely used in the field of information processing. One-dimension and two-dimension quantum wavelet transforms have been investigated as important tool algorithms. However, three-dimensional quantum wavelet transforms have not been reported. This paper proposes a multi-level three-dimensional quantum wavelet transform theory to implement the wavelet transform for quantum videos. Then, we construct the iterative formulas for the multi-level three-dimensional Haar and Daubechies D4 quantum wavelet transforms, respectively. Next, we design quantum circuits of the two wavelet transforms using iterative methods. Complexity analysis shows that the proposed wavelet transforms offer exponential speed-up over their classical counterparts. Finally, the proposed quantum wavelet transforms are selected to realize quantum video compression as a primary application. Simulation results reveal that the proposed wavelet transforms have better compression performance for quantum videos than two-dimension quantum wavelet transforms.

Keywords wavelet transform wavelet video coding quantum wavelet transform quantum information processing quantum image processing

Corresponding Author(s): Haisheng LI

Just Accepted Date: 20 September 2022 Issue Date: 29 December 2022

Cite this article:

Haisheng LI,Guiqiong LI,Haiying XIA. Three-dimensional quantum wavelet transforms[J]. Front. Comput. Sci., 2023, 17(5): 175905.

URL:

https://academic.hep.com.cn/fcs/EN/10.1007/s11704-022-1639-y
https://academic.hep.com.cn/fcs/EN/Y2023/V17/I5/175905

Fig.1 Notations for some basic gates

Fig.2 Quantum controlled-NOT gates. Control quantum bits can be 0 (indicated by white dots) or 1 (indicated by black dots)

Fig.3 The implementation circuits of

P 2 n ? 1, 2

and

P 2, 2 n ? 1

. Circuits in the dashed boxes in (a) and (b) realize

P 2 n ? 2, 2

and

P 2, 2 n ? 2

, respectively

Fig.4 The circuits for the single-level 1D-HQWT and its inverse. (a)

W 2 n H

; (b)

(W 2 n H) ? 1

Fig.5 The circuits for the single-level 1D-D4QWT and its inverse. (a)

D 2 n p

; (b)

F 2 n

; (c)

(D 2 n p) ? 1

; (d)

F 2 n ? 1

Fig.6 The decomposition scheme of the classical 2-level 3D wavelet transform

Fig.7 Circuits for

T n, m, h j

and

R n, m, h j

and their inverses. (a)

T n, m, h j

; (b)

R n, m, h j

; (c)

(T n, m, h j) ? 1

; (d)

(R n, m, h j) ? 1

Fig.8 Circuits for the

i

-level 3D-QWT. (a)

D n, m, h j

; (b)

L n, m, h j

; (c) The i-level 3D-QWT

Fig.9 The implementation circuits for the multi-level 3D-HQWT. (a)

H n, m, h i, i ≥ 2

; (b)

H n ? i + 1, m ? i + 1, h ? i + 1 1

Fig.10 The implementation circuits for the inverse of the multi-level 3D-HQWT. (a)

(H n, m, h i) ? 1, i ≥ 2

; (b)

(H n ? i + 1, m ? i + 1, h ? i + 1 1) ? 1

Fig.11 The circuit for

H 4, 5, 5 3

. The circuit in the dashed box implements

H 3, 4, 4 2

Fig.12 The implementation circuits for the multi-level 3D-D4QWT. (a)

F n, m, h i

with

i ≥ 2

; (b)

F n ? i + 1, m ? i + 1, h ? i + 1 1

Fig.13 The implementation circuits for the inverse of the multi-level 3D-D4QWT. (a)

(F n, m, h i) ? 1

with

i ≥ 2

; (b)

(F n ? i + 1, m ? i + 1, h ? i + 1 1) ? 1

Fig.14 The circuit for

F 4, 5, 5 3

. The circuit in the dashed box implements

F 3, 4, 4 2

Fig.15 The circuits for 3-level 1D-HQWTs

H 43

and

H 53

. (a)

H 43

; (b)

H 53

Fig.16 The circuit for the 3-level 2D-HQWT

H 4, 5 3

Fig.17 Simulation results of the first two levels of the 3D-HQWT. (a) A

64 × 64 × 8

original video; (b) the transformed video for the 1-level 3D-HQWT; (c) the transformed video for the 2-level 3D-HQWT

Fig.18 Simulation results of the first two levels of the 3D-D4QWT. (a) A

64 × 64 × 8

original video; (b) the transformed video for the 1-level 3D-D4QWT; (c) the transformed video for the 2-level 3D-D4QWT

Level	$n o r m (V 1 ? W 1)$	$n o r m (V 2 ? Λ 26, 26, 25)$	$n o r m (V 3 ? W 3)$	$n o r m (V 4 ? Λ 26, 26, 25)$
1	$7 × 10 ? 12$	$3.3 × 10 ? 11$	$3.19 × 10 ? 9$	$3.47 × 10 ? 11$
2	$3.2 × 10 ? 11$	$5.9 × 10 ? 11$	$5.72 × 10 ? 9$	$8.16 × 10 ? 11$
3	$4.05 × 10 ? 10$	$3.34 × 10 ? 10$	$8.24 × 10 ? 9$	$2.371 × 10 ? 10$
4	$1.527 × 10 ? 9$	$1.636 × 10 ? 9$	$1.238 × 10 ? 8$	$5.515 × 10 ? 10$
5	$7.746 × 10 ? 9$	$7.876 × 10 ? 9$	?	?

Tab.1 Comparisons of the proposed QWTs and the corresponding classical wavelet transforms

Fig.19 An

8 × 8 × 8

original video

Fig.20 Quantum circuit of an

8 × 8 × 8

video with 2-level 3D-HQWT. (a) The first part of the circuit; (b) the second part of the circuit

Fig.21 Probability histogram of an

8 × 8 × 8

video with 2-level 3D-HQWT after 8192 quantum measurements

Fig.22 Quantum circuit of an

8 × 8 × 8

video with 2-level 3D-D4QWT. The symbols circuit401, circuit393, and circuit397 represent the NOT, S0, and S1 gates, respectively. (a) The first part of the circuit; (b) the second part of the circuit; (c) the third part of the circuit

Fig.23 Probability histogram of an

8 × 8 × 8

video with 2-level 3D-D4QWT after 8192 quantum measurements

Fig.24 Comparisons of QVC-3D-HQWT and QIC2DHQWT. HW1=

i

and HW2=

i

(

i = 1, 2, 3, 4, 5

) denote the

i

-level 3D-HQWT and 2D-HQWT, respectively

Fig.25 Comparisons of QVC-3D-D4QWT and QIC2DD4QWT. DW1=

k

and DW2=

k

(

k = 1, 2, 3, 4

) denote the

k

-level 3D-D4QWT and 2D-D4QWT, respectively

QWTs	3D-HQWT	2D-HQWT	3D-D4QWT	2D-D4QWT
PSNR	30.39	30.40	30.14	30.81
QCR	29.08	12.94	22.06	11.61
Threshold	$λ 6 H$	$β 4 H$	$λ 1 D$	$β 2 D$
Level	5	5	4	4

Tab.2 QCR and PSNR for simulation results

Fig.26 Simulation results for QVC-3D-HQWT and QVC-3D-D4QWT. (a) An original video; (b) the compressed video by QVC-3D-HQWT; (c) the compressed video by QVC-3D-D4QWT

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