The multi-level and multi-dimensional quantum wavelet packet transforms.

The classical wavelet packet transform has been widely applied in the information processing field. It implies that the quantum wavelet packet transform (QWPT) can play an important role in quantum information processing. In this paper, we design quantum circuits of a generalized tensor product (GTP) and a perfect shuffle permutation (PSP). Next, we propose multi-level and multi-dimensional (1D, 2D and 3D) QWPTs, including a Haar QWPT (HQWPT), a 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, we analyze the quantum costs and the time complexities of our proposed QWPTs and obtain precise results. The time complexities of HQWPTs is at most 6 on 2n elements, which illustrates high-efficiency of the proposed QWPTs. Simulation experiments demonstrate that the proposed QWPTs are correct and effective.

Introduction

With the rapid development in the fields of optical imaging, Internet technology, high performance calculation etc., the amount of data is increasing explosively, so that it is necessary to find new ways to store and process information. Quantum information processing (QIP)[1] as new technology of information processing, offers a potential solution to store and process massive visual data efficiently. QIP has two outstanding merits: (1) the unique computing performance of quantum coherence, entanglement and superposition [1], and (2) quantum storage capacity increasing exponentially. Models of quantum image representation[2-8] have displayed the enormous storage capacity of QIP. Other popular quantum algorithms, such as the Shor’s discrete logarithms and integer-factoring algorithms[9], the Deutsch’s parallel computing algorithm[10] and the Grover’s quadratic speed up algorithm[11], have further shown that QIP is more efficient than its classical counterparts. In addition, many algorithms of QIP emerge continually, and these algorithms include quantum geometric transformation[12-14], quantum image encryption and decryption algorithms[15,16], quantum watermarking[17], quantum image compression[6], quantum edge detection[18], and quantum image filtering[19].

The classical wavelet packet transform (WPT) has been widely spread to the information processing field for image coding[20], pattern matching[21] and fractional brownian motion decorrelation[22]. It indicates that the quantum wavelet packet transform (QWPT) plays an important role in QIP. Unfortunately, the research on QWPT is rare and still preliminary. For example, two important QWPTs, namely the Haar QWPT (HQWPT) and the D4 QWPT (DQWPT) proposed in[23-26], are still single level quantum wavelet transforms. Up to now, we have not yet found any implementation of a multi-level and multi-dimensional QWPT. Therefore, we believe that QWPTs deserve further research.

In this paper, we introduce the generalized tensor product (GTP) and the perfect shuffle permutation (PSP), and design quantum circuits for them. Then, we propose the iterations and implementation circuits of the multi-level and multi-dimensional QWPT and inverse QWPT (IQWPT). QWPTs and the inverse QWPTs being considered include HQWPT, DQWPT based on a periodization extension, the inverse HQWPT (IHQWPT), the inverse DQWPT (IDQWPT). In addition, we analyze the quantum costs and time complexities of the proposed circuits and prove that the multi-level and multi-dimensional HQWPT can be implemented with a complexity of O(1). Simulation experiments demonstrate that the proposed QWPTs are correct and effective.

The contributions of this paper are listed as follows.

We analyze precisely the complexities of the simulated networks of controlled NOT gates with multi-control qubits. Comparing with the methods proposed in the reference[27], our proposed simulated networks are reduced by 50% approximately.

We design the simplified circuits of the PSP and reduce time complexity to 6 for 2 elements.

We present the multi-level and multi-dimensional QWPTs, including HQWPT, IHQWPT, DQWPT and IDQPT for the first time, and prove the correctness by theoretical derivations and simulation experiments.

We design the circuits of the multi-level and multi-dimensional HQWPT with the complexity O(1), which has the overwhelming advantage over the classic Haar WPT.

The Quantum Implementation of GTP

Let A be an n × n matrix and B be an m × m matrix, then the tensor product is an mn × mn block matrix in the following equation,

Thus, the tensor product of quantum states are defined as the tensor product of matrices: , which is also written simply as or .

Then, n fold tensor product is abbreviated as . Similarly, the abbreviation of is .

A larger vector space can be formed by putting vector spaces together. For instance, suppose that is a basic state in a 2 dimensional Hilbert space for i = 0, 1, …, 2 − 1, the state consists of the tensor products of the n computation basis states:where and i1, i2, …, i ∈ {0, 1}. Its dual state is

There are some base gates and their corresponding symbols shown in Fig. 1. In the figure, the identity (I2), Hadamard (H), Pauli-X (X) and Swap gates are well-known and can be found in the reference[28]. The 2 × 2 identity matrix denotes the circuit of n qubits. V and V+ are two specific examples of U gates where U corresponds to a unitary matrix and

Figure 1

Notations for some base gates with their corresponding symbols.

A controlled gate is one of the most useful gates in quantum computing, and we define two controlled gates of (n + m)-qubits.

Definition 1.

Let be a 2 × 2 unitary matrix, be a 2 × 2 identity matrix. Then, controlled gates and with n control qubits and m target qubits are defined bywhere and are the basic states in a 2 dimensional Hilbert space shown in Eq. (2), and j ∈ {0, 1, …, 2 − 1}. The Notations of and are shown in (a) and (b) of Fig. 2. Furthermore, and are called Toffoli gates.

Figure 2

The (n + m) qubit controlled gates and the gate. The abbreviation notations are in the right parts of (a,b). The dashed box 1 and 2 in (c) implement and where j = j j … j1 and j, j, … j1, j0 ∈ {0, 1}.

Definition 2.

An (n + m) qubit controlled gate with n control qubits is named as an gate, when the X gate is in the target qubit of the controlled gate. An instance of an gate is shown in (c) of Fig. 2. In addition, the four gates shown in Fig. 3 are called controlled-NOT gates.

Figure 3

The four gates. The numbers 1 and 0 can be replaced by black and white points on control qubits.

A Swap gate can be simulated by three gates, that is, .

Next, we introduce a perfect shuffle permutation. Let P be the mn × mn matrix of a perfect shuffle permutation, then P satisfies that (P) = δδ where k = vn + z, l = v′m + z′, 0 ≤ v, z′ < m, 0 ≤ v′, z < n, δ is the Kronecker delta function, that is, δ = 0 if x ≠ y, otherwise δ = 1. Therefore, P shuffles n packs of m cards into m packs of n cards.

As a useful tool for wavelet transforms, the GTP is defined as follows[29]. Suppose that and are two sets of matrices, where A is an n × n matrix, 0 ≤ i < m, and B is an m × m matrix, 0 ≤ j < n. Then, the generalized tensor product is an mn × mn matrix and can be calculated bywhere and are block diagonal matrices.

Definition 3.

Let and be two sets of matrices where A and D are n × n matrices. Then, the generalized product is defined as .

Definition 4.

The transpose, conjugate transpose and inverse of the matrix set are defined as follows:where (A), (A)+ and (A)−1 denote the transpose, conjugate transpose and inverse of matrix A, respectively.

The following equations hold by using equation (7) and definitions 3 and 4.

Let and be two sets of matrices containing m matrices with size n × n, and be two sets of matrices containing n matrices with size m × m, and I and I be m × m and n × n identity matrices, respectively. Then, the following equation holds[24]:and implies

Furthermore, calculating by the definition of a GTP, we can implement the following four GTPs using controlled gates:

The Complexity Analysis of Quantum Circuits

The complexity analysis of quantsssum circuits

Since a quantum circuit can be simulated by basic operations referring to single-qubit gates, controlled-NOT gates, controlled-V and controlled-V+ gates[12,27,28,30], we introduce some definitions and lemmas. Furthermore, and are the symbols of round down and round up respectively, which are used in the following definitions and lemmas.

Definition 5.

The quantum cost of a quantum circuit can be regarded as the total number of basic operations which simulate the circuit, marked by .

Definition 6.

The time complexity of a quantum circuit is defined by the total number of time steps. In a time step, only one basic operation is executed serially, but multiple ones can be performed in parallel. It is marked by .

Lemma 1.

When n ≥ 6 and , an gate can be simulated by a network consisting of 2(m − 1) Toffoli gates and a basic operation.

For instance, and gates can be simulated by 2(m − 1) Toffoli gates and a basic operation, respectively. The form of the network is shown in (a) and (b) of Fig. 4.

Figure 4

The controlled gate illustrated for n = 10 and m ∈ {5, 6}.

Lemma 2.

For any n ≥ 6, and m ∈ {r + 1, r + 2, …, n − 2}, an gare can be simulated by two gates and two gates.

For instance, the simulated networks of and gates are shown in (c) and (d) of Fig. 4.

Lemma 3.

When n ≥ 5 and , an gate can be simulated by a network consisting of 4(m − 2) Toffoli gates.

For instance, the simulated networks of and gates are shown in Fig. 5.

Figure 5

The controlled gate illustrated for n = 9 and m = 5.

Lemma 4.

For any SU(2) matrix D, there exist SU(2) matrices A, B, and C such that ABC = I2 and AXBXC = D, and the gates and can be simulated by networks of the form shown in (a) and (b) Fig. 6. Here, SU(2) is the Lie group of 2 × 2 unitary matrices with determinant 1.

Figure 6

The simulated networks of the gates and with n ≥ 2.

More details of Lemmas 2, 3 and 4 are described in the reference[27]. Next, we derive the following corollaries.

Corollary 1.

For any n ≥ 7, and m ∈ {r + 2, r + 3, …, n − 2}, an gate can be simulated by 4(m − 1) Toffoli gates and four basic operations.

Proof.

Applying Lemma 2, an gate can be simulated by two gates and two gates. Noting that and , we apply lemma 1 so that the corollary holds.

Corollary 2.

For any n ≥ 6 and , an gate can be simulated by (4r − 2) Toffoli gates and two basic operations when n is even, and 4(r − 1) Toffoli gates when n is odd.

When n is odd, . Then, applying Lemma 3, we have that an gate can be simulated by a network consisting of 4(r − 1) Toffoli gates.

When n is even, by applying Lemma 2, it is derived that an gate can be simulated by two gates and two Toffoli gates. Then, by applying Lemma 1, it is proved that one can use (4r − 2) Toffoli gates and two basic operations to simulate the gate.

From lemma 4, the following corollary holds.

Corollary 3.

For any SU(2) matrix D, there exist SU(2) matrices A, B, and C such that ABC = I2 and AXBXC = D, and the gates and can be simulated by networks of the form shown in (c) and (d) of Fig. 6.

To analyze the complexities of the gates and , we define three matrices:

Lemma 5.

Let , i ∈ {0, 1, 2, …, n}, E0 = Φ(δ)R(δ). Then, the gates and can be simulated by networks of the form shown in Fig. 7.

Figure 7

The simulated networks of the gates and .

Note that

Then, we have that

Similarly,

Therefore, we have the simulated networks of the gates and as shown in Fig. 7.

Lemma 6.

The gates and can be simulated by networks of the form shown in Fig. 8.

Figure 8

The simulated networks of the gates and .

Due to and , the conclusion is obvious.

(i = 0, 1, 2, 3) can be simulated by five basic operations shown in Fig. 9, i.e., .

Figure 9

The simulated networks of the gates (i = 0, 1, 2, 3).

Similarly, . Therefore, the complexity of Toffoli gates is 5. Thus, we obtain the complexity of and as described in theorem 1 below.

Theorem 1.

For any n ≥ 7, the gates and can be simulated by (3.5n2 − 13n − 4) Toffoli gates and 7n − 4 basic operations when n is even, and by (3.5n2 − 12n − 5.5) Toffoli gates and 7n − 3 basic operations when n is odd.

Let δ = π/2, D = R(−π)R(π), A = R(−π)R(π), B = R(−π/2)R(π/2) and C = R(π/2). Then, D, A, B, C ∈ SU(2), ABC = I2, AXBXC = D and Φ(δ)D = X.

Note that . Then,

From lemma 4 and corollary 3, we obtain

By lemma 5 and lemma 6, can be computed by

Therefore,where and is a Toffoli gate.

Applying lemma 1, corollary 1 and corollary 2, we obtain

Similarly, we obtain that .

Comparing with the methods proposed in[27], the complexities of our proposed simulated networks of these gates are reduced by 50% approximately.

The Quantum Circuits of PSP

The perfect shuffle permutation and can be expressed aswhere P2,2 is a Swap gate, and their implementation circuits are shown in Fig. 10.

Figure 10

The implement circuits of and . The dotted boxes in (a,b) are the circuits of and , respectively.

Applying and to the state , we have

Let , we have that and

Therefore, we conclude that and design quantum circuits shown in Fig. 11.

Figure 11

The simplified quantum circuits of and .

The costs of the circuits of and are

By parallel computing, we redesign the circuits of and shown in Fig. 12 and calculate time complexities byi.e., complexities of and are O(1).

Figure 12

The parallel quantum circuits of and .

The iterations of and are given by

Then, we obtain

Therefore, we design the simplified circuits of and as shown in Fig. 13.

Figure 13

The simplified circuits of and . The rights of (a,b) correspond to the abbreviation notation of and , respectively.

The complexities of and are

The reason that the abbreviation notations in Fig. 13 are the same except for the positions of black boxes is due to the fact that the circuit in Fig. 13(b) consists of the gates in Fig. 13(a) but rearranged in reverse order. We also adopt similar abbreviation notations to denote the circuits that are composed of the same quantum gates with reverse order in the following sections.

The iterations of and are given by

According to (30), we design the implementation circuits of and in Fig. 14.

Figure 14

The quantum circuits of and . The dotted boxes in (a,b) are the implement circuits of and , respectively.

The complicities of the circuits in Fig. 14 are

The Implementation of QWPT

Let be a wavelet kernel matrix. Then, the (k + 1)-th iteration of a discrete wavelet packet transform is defined bywhere j = 1, …, k and is a matrix with 2 blocks of on the main diagonal and zeros elsewhere.

The following equationscan be derived by (32).

Sincethe iteration equation of the QWPT is given bywith the initial value and the implementation circuit shown in (a) of Fig. 15.

Figure 15

The implementation circuits of and .

Similarly, the inverse of iswith the initial value and the implementation circuit of shown in (b) of Fig. 15.

Next, we describe the implementations of the Haar QWPT (HQWPT) and the D4 QWPT (DQWPT) in detail.

The implementation of HQWPT

Substituting the kernel matrix into equations (35) and (36), the (k + 1)-th iteration of HQWPT and its inverse arewith the initial values

The quantum circuits of and (1 ≤ k < n − 1) are designed in Fig. 16.

Figure 16

The implementation circuits of and (1 ≤ k < n − 1). The dashed box 1 and box 3 implement and , respectively.

Since and with 1 ≤ k < n − 2, the quantum circuit of , and can be simplified and shown in Fig. 17.

Figure 17

The simplified circuits of 1 ≤ k < n − 2, and .

Similarly, the quantum circuits of the inverses of , and can be designed as shown in Fig. 18.

Figure 18

The simplified circuits of 1 ≤ k < n − 2, and .

The costs of HQWPT arewhere 1 ≤ k < n − 2. Since and , the time complexity of the HQWPT is O(1).

The implementation of DQWPT

The kernel matrix of the D4 wavelet transform is defined by the reference[31]where , , and .

and can be rewritten towhereand the implementation circuits shown in Fig. 19.

Figure 19

The quantum circuits of the kernel matrix of the D4 wavelet transform.

In order to implement a multi-level DQWPT based on the periodization extension, a single-level DQWPT and its inverse are given by:

The implement circuits of the above DQWPT are shown in Fig. 20. Substituting the kernel matrix with in (35) and (36), we obtain that the (k + 1)-th iterations of the DQWPT and its inverse based on the periodization extension arewith the initial values , , 1 ≤ k < n − 1 and their implementation circuits shown in Fig. 21.

Figure 20

The quantum circuits of the single-level DQWPT and its inverse based on the periodization extension.

Figure 21

The quantum circuits of and . The dashed boxes in (a,b) implement and , respectively.

Using , the quantum circuit of and can be simplified and shown in Fig. 22.

Figure 22

The simplified circuits of and with 1 ≤ k < n − 1.

We analyze the complexity of the above DQWPT and suppose .

From Figs 19 and 20, we calculate the complexity of by

Applying lemma 1, corollary 1, corollary 2 and theorem 1, we obtain

We calculate the quantum cost of by

Let the time complexity of DQWPT is

For instance,and

The costs of and are

The 2D and 3D QWPTs

Firstly, we briefly describe NASS to represent 2D images and 3D videos. The NASS state of an image can be represented bywhere and are the X-axis and Y-axis of the image, represents the color of the pixel in the coordinate , and n = m + k.

The NASS state of a video can represented bywhere , and are the X-axis, Y-axis and time-axis of a video, and n = m + k + h.

More details are shown in our previous work[6]. For instance, the NASS staterepresents the color image of 8 × 4 (height multiplies weight) as shown in (a) of Fig. 23.

Figure 23

The image and the video.

The NASS staterepresents the video with four frames as shown in (b) of Fig. 23, where each frame is a 4 × 2 image.

The same string can have different meanings corresponding to different data types in classic computers. For instance, a binary string 0100001 can represent a char ‘A’ or a number 65. Similarly, using the circuit in[6], we can store an image (shown in (a) of Fig. 23) or a video (shown in (b) of Fig. 23) in the following state

Meanwhile, the priori knowledge ‘x3, y2’ or ‘x2, y1, t2’ is equivalent to a data type, implying an image or a video stored in the state .

A natural image with size of 2 × 2 can be expressed as an angle matrixwhere θ is the color information of the pixel on the coordinate (x, y) and an example is shown in Fig. 23.

Thus, the 2D wavelet transform on is defined aswhere and are 2 × 2 and 2 × 2 wavelet transforms, respectively

An image can be stored in the state NASS in (50) by using a quantum circuit in the literature[6]. Suppose that the function is equivalent to the quantum circuit implementing the storage of the image , that is,where is the row vector of and 0 ≤ j ≤ 2 − 1.

Applying the function on , the result is

Using the perfect shuffle permutation , we obtain

Then, we have

Then, the 2D QWPT of is given by

A video of 2 frames of size 2 × 2 corresponds to the following angle matrix.where the angle matrix is the k-th frame.

We firstly define the following DWPTs: , and .with the row vectorswhere and are the elements of the matrices and on the position (x, y), respectively.

Next, the 3D DFPT of can be defined as

Similarly, we utilize the equivalent function of the quantum circuit to create the NASS state of where the row vector u is shown in equation (65).

Applying the function on , and respectively, we have the following three equations.

Therefore, we derive the 3D QWPT of

Substituting our proposed 1D QWPT into equations (62) and (69), we obtain 2D HQWPT, 2D DQWPT, 3D HQWPT and 3D DQWPT. Furthermore, their circuits can be designed in Figs 24 and 25.

Figure 24

The quantum circuits of the 2D QWPT and IQWPT with 1 ≤ k ≤ min(m, n) − 1 in (e,f), 1 ≤ k ≤ min(m, n) − 2 in (f,h).

Figure 25

The quantum circuits of the 2D QWPT and IQWPT with 1 ≤ k ≤ min(m, n, p) − 1 in (e,f), 1 ≤ k ≤ min(m, n, p) − 2 in (f,h).

Simulation Experiments

In the absence of a quantum computer to implement our proposed QWPTs, experiments of quantum signals are simulated on a classical computer. The quantum signals are stored in quantum states (i.e., column vectors) and the QWPTs are implemented using unitary matrices in Matlab (the R2010bversion).

Simulation experiments of the 1D HQWPT and DQWPT

Consider a quantum stateas an input signal of the QWT, where v = d(k/2048), k = 0, …, 2047, and .

For simply, we can take a vectoras the input signal of simulation experiments, which is according with the state without the normalized item.

For convenience, let the single-level HQWPT and DQWPT be

Applying multi-level HQWPT and multi-level DQWPT to the input signal S in Eq. (71), the simulation results of the first 3 levels are shown in Fig. 26 with multi-windows. Table 1 shows the comparison of simulation experiments of our proposed QWPT and the function of the WPT in Matlab using the 2-norm function . The symbols in this table are listed as follows:

Figure 26

The simulation results of the first 3 levels of HQWPT and DQWPT. The left number i refers to i-level QWT with 1 ≤ i ≤ 10, and i = 0 refers to the input signal.

Table 1

The simulation results of the HQWT, IHQWT and HWT.

level	norm(M1k−M5k)×10^13	norm(S−M2k)×10^13	norm(M3k−M6k)×10^10	norm(S−M4k)×10^13
1	0	0.0211	0.0057	0.0257
2	0.0228	0.0583	0.0128	0.0511
3	0.0252	0.0643	0.0230	0.0798
4	0.0361	0.0619	0.0374	0.1006
5	0.0311	0.0716	0.0574	0.1371
6	0.0693	0.0736	0.0842	0.1662
7	0.0505	0.1118	0.1168	0.2402
8	0.0960	0.1518	0.1564	0.2575
9	0.1325	0.1874	0.1963	0.3040
10	0.1982	0.2899	0.1560	0.3623
11	0.2497	0.3981	—	—

The function in Matlab performs a (k + 1)-level HWPT to return a wavelet packet tree. Next, we get the coefficients of the nodes of the wavelet packet tree using the function to construct a vector . similarly, we obtain a a vector of a (k + 1)-level DWPT based on the periodization extension by the and .

Simulation experiments of the 2D HQWPT and DQWPT

An angle matrix Λ is given bywhere A is the 128 × 128 matrix of the gray-scale image shown in Fig. 27(a), and C is a constant corresponding to the image.

Figure 27

The simulation results of the first 2 levels of HQWPT and DQWPT.

The NASS state can be regarded as a column vector, where the function is defined in equation (57). Applying the k + 1 level 2D HQWPT and DQWPT on the image A, respectively, the results arewhere is the inverse function of , which converts a column vector into a 2-dimension matrix.

The simulation results are shown in Fig. 27 and Table 2. The rest symbols in Table 2 are: , . Similarly with the 1D HQWPT and DQWPT, matrices and are created using the functions , and , respectively.

Table 2

The simulation results of the 2D HQWT, IHQWT and HWT.

level	norm(Q1k−Q5k)×10^9	norm(|ψ_2〉−Q2k)×10^13	norm(Q3k−Q6k)×10^8	norm(|ψ_2〉−Q4k)×10^14
1	0.0011	0.0041	0.0669	0.0427
2	0.0022	0.0093	0.0128	0.0880
3	0.0026	0.0104	0.2641	0.1837
4	0.0099	0.0125	0.3917	0.2904
5	0.0218	0.0242	0.5851	0.6436
6	0.0679	0.0518	0.9105	0.8180
7	0.1218	0.1079	—	—

Simulation experiments of the 3D HQWPT and DQWPT

An angle matrix Λ is given bywhere V is the 64 × 64 × 4 matrix of the video shown in (a) of Fig. 28, and C is a constant corresponding to the video.

Figure 28

The simulation results of the first 2 levels of the 3D HQWPT and DQWPT.

The NASS state can be regarded as a column vector, where the function is defined in equation (67). Applying the k + 1 level 3D HQWPT and DQWPT on the video V, respectively, the results arewhere converts a column vector into a 3-dimension matrix.

The simulation results are shown in Fig. 28 and Table 3. Since there are no functions of the 3D WPT, we realize wt3 in (66) using the functions and and note and as results of 3D HWPT and DWPT, respectively. The rest symbols in Table 2 are and .

Table 3

The simulation results of the 3D HQWT, IHQWT and HWT.

level	norm(V1k−V5k)×10^10	norm(|ψ_3〉−V2k)×10^14	norm(V3k−V6k)×10^8	norm(|ψ_3〉−V4k)×10^14
1	0.0164	0.0488	0.1787	0.0714
2	0.0702	0.1449	0.3328	0.2005
3	0.2171	0.2044	—	—

Analyzing the above simulation experiments, we conclude that our proposed HQWPT, IHQWPT, DQWPT and IDQWPT can implement decompositions and reconstructions of the Haar wavelet and D4 wavelet, respectively. The simulation results of our proposed HQWPT and DQWPT, which are equal to the corresponding WPTs without consideration of truncation error on machine computing, show our proposed QWPTs are correct.

Conclusion and Future Works

This article has constructed the iteration equations of multi-level and multi-dimensional QWPTs by GTP and PSP. The iteration equations include HQWPT, DQWPT based on the periodization extension and their inverse transforms for the first time, which ensure the theoretical correctness of our proposed QWPTs. Next, we have designed circuits of the proposed QWPTs. The precise analysis of the quantum costs and the time complexities of circuits prove that our proposed QWPTs are of high-efficiency. For instance, the time complexities of the multi-level HQWPT and DQWPT at most are 6 and (5n3 + O(n2)) on 2, respectively. In contrast, the classical fast WPTs need O(n2) basic operations to implement the discrete wavelet transform[21,32]. Thus, our proposed QWPT can exponentially speed up the computation of the wavelet transform in comparison to the one on a classical computer. The simulation results show that our proposed QWTs are correct and effective. In summary, the proposed QWPTS and IQWPTs can implement effective decompositions and reconstructions of 1D signals, 2D images and 3D vedio, respectively. Therefore, the article provide a feasible scheme for the WPT to be applied in QIP.

Studies of quantum wavelet packet are still in their infancy. Multi-level and multi-dimension wavelet transforms play an important role in classical image and signal processing, therefore, their quantum versions will be significant and core tool algorithms for quantum image and signal processing. Our future works are how to use these wavelet transforms to implement some complex operations, such as quantum image and signal compression, and quantum image and signal denoising.