CN107153632B - Method for realizing quantum circuit design by quantum Haar wavelet transform - Google Patents

Method for realizing quantum circuit design by quantum Haar wavelet transform Download PDF

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CN107153632B
CN107153632B CN201710324420.9A CN201710324420A CN107153632B CN 107153632 B CN107153632 B CN 107153632B CN 201710324420 A CN201710324420 A CN 201710324420A CN 107153632 B CN107153632 B CN 107153632B
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黎海生
夏海英
宋树祥
范萍
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Guangxi Normal University
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Abstract

The invention provides a method for realizing quantum circuit design by quantum Haar wavelet transform, belonging to the field of quantum information processing, which is used for perfecting and improving the existing quantum Fourier transform realization technology, and 2 realization circuits of multilayer quantum Haar wavelet transform and 2 realization circuits of multilayer quantum Haar wavelet inverse transform are respectively constructed by using expanded tensor products and basic quantum bit gates (including quantum bit controlled gates and single quantum bit gates). From the line complexity analysis of the realization of the quantum Haar wavelet transform and the quantum Haar wavelet inverse transform, it can be known that for one 2nThe complexity of the lines of the data set of the individual elements, the 2 multi-layer quantum Haar wavelet transforms and the 2 multi-layer quantum Haar wavelet inverse transforms is theta (n)2) This is not achievable with other classical fast Haar wavelet transforms. The method is suitable for the algorithm fields of compression, denoising, encryption, decryption and the like of images in many practical information processing applications, and has great significance for quantum computing theory perfection and application popularization.

Description

Method for realizing quantum circuit design by quantum Haar wavelet transform
Technical Field
The invention relates to the field of quantum information processing, in particular to a method for realizing quantum circuit design by quantum Haar wavelet transform.
Background
Quantum computing is a product combining quantum mechanics and computer science, and the parallelism, the superposition and the uncertainty of measurement of the quantum computing are the basis of the quantum computer superior to a classical computer.
The quantum wavelet transform is a core algorithm of quantum information processing. The quantum Haar wavelet transform is a compact, second-order and orthogonal wavelet transform and has wide application in the field of information processing, for example, the quantum Haar wavelet transform plays an important role in algorithms such as image coding, edge detection, image watermarking and the like.
In classical calculations, an information element is represented by a Bit (Bit), which has only two states: the 0 state or the 1 state. In quantum computing, an information element is represented by a Qubit, which has two fundamental quantum states |0 > and |1 >, which are referred to as the ground State (Basis State). A qubit can be a linear combination of two ground states, often referred to as a Superposition (superpositioning), which can be expressed as | ψ > ═ a |0 > + b |1 >. Wherein a and b are two complex numbers satisfying | a tint2+|b|21 and is therefore also referred to as a probability amplitude. The ground states |0 > and |1 > can be represented as vectors:
Figure GDA0002979553400000011
their dual vectors can be represented as: < 0| ═ 10, < 1| ═ 01.
Tensor product is a method of combining small vector spaces together to form a larger vector space, using symbols
Figure GDA0002979553400000012
It has the following meanings:
suppose U is n × n and V is m × m two complex matrices
Figure GDA0002979553400000021
Then
Figure GDA0002979553400000022
Assume that the two unitary matrix sets are:
Figure GDA0002979553400000023
and
Figure GDA0002979553400000024
in which there are m n x n matrices,
Figure GDA0002979553400000025
there are n m by m matrices. The expanded tensor product is a matrix of mn × mn
Figure GDA0002979553400000026
Wherein
Figure GDA0002979553400000027
When in use
Figure GDA0002979553400000028
Is the same for each matrix in (A)iWhen it is A
Figure GDA0002979553400000029
Can be written as
Figure GDA00029795534000000210
If at the same time
Figure GDA00029795534000000211
Each matrix in (a) is the same as (B)iB, the tensor product of the expansion at this time
Figure GDA00029795534000000212
Degenerating into ordinary tensor product
Figure GDA00029795534000000213
A quantum wire may be composed of a sequence of qubit gates, each line representing a connection of the quantum wire in the representation of the quantum wire, the sequence of execution of the quantum wire being from left to right. The qubit gates can be conveniently represented in matrix form, and the single qubit gate can be represented by a 2 x 2 unitary matrix U, i.e. U+U is I, wherein U+Is a conjugate transpose of UMatrix, I is a unit matrix. The names, symbols and corresponding matrix representations of some basic qubit gates are shown in fig. 1.
Pn,mIs a uniform shuffling permutation matrix of mn by mn, (P)n,m)k,l＝δv,z'δz,v'Where k is vn + z, l is v'm + z', 0. ltoreq. v, z '< m, 0. ltoreq. v', z < n, when x.ltoreq.y, δx,yIf it is 0, n isx,y＝1。
Figure GDA00029795534000000214
And
Figure GDA00029795534000000215
are two special uniform shuffling permutation matrices whose recursive equation is:
Figure GDA0002979553400000031
Figure GDA0002979553400000032
where Swap is the switch gate shown in figure 1,
Figure GDA0002979553400000033
as shown in figure 2,
Figure GDA0002979553400000034
the quantum implementation line of (a) is shown in fig. 3.
The complexity of the electronic circuit design realized by the existing classical Haar wavelet transform is theta (n 2)n) Is complex and can not well meet the social requirements. There is a need to devise less complex ways of implementing electronic circuit designs.
Disclosure of Invention
The invention provides a method for realizing quantum circuit design by quantum Haar wavelet transform, which solves the problem of high complexity of electronic circuit design realized by the conventional classical Haar wavelet transform
The invention solves the problems through the following technical scheme:
a method for realizing quantum circuit design by quantum Haar wavelet transform is disclosed, wherein quantum computation is combined with classical Haar wavelet transform technology to obtain quantum Haar wavelet transform; designing quantum circuits of 2 quantum Haar wavelet transforms and 2 quantum Haar wavelet inverse transforms according to the expanded tensor product operation principle by using the quantum Haar wavelet transforms;
2 quantum lines of the quantum Haar wavelet transform are respectively a quantum line of the first multilayer quantum Haar wavelet transform and a quantum line of the second multilayer quantum Haar wavelet transform;
the quantum lines of the 2 quantum Haar wavelet inverse transforms are the quantum lines of the first multilayer quantum Haar wavelet inverse transform and the quantum lines of the second multilayer quantum Haar wavelet inverse transform, respectively.
In the foregoing solution, it is preferable that the design implementation process of the quantum line of the first multilayer quantum Haar wavelet transform is as follows:
is provided with
Figure GDA0002979553400000035
Is k layer 2nA Haar wavelet transform of the elements,
Figure GDA0002979553400000036
is a k-1 layer 2n-1Elemental Haar wavelet transform, then the classical multi-layer quantum Haar transform can be defined as:
Figure GDA0002979553400000037
wherein
Figure GDA0002979553400000038
Is a uniform shuffling permutation matrix, H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000039
is I2The product of the n tensors.
According to the principle of the expanded tensor product operation, obtaining
Figure GDA00029795534000000310
Substituting the formula into the formula (1), and designing a first iteration formula of multilayer quantum Haar transformation as follows:
Figure GDA0002979553400000041
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000042
is the sign of the operation of the tensor product,
Figure GDA0002979553400000043
is I2The product of the n tensors of (a),
Figure GDA0002979553400000044
is a matrix of uniform shuffling permutations,
Figure GDA0002979553400000045
is k layer 2nA Haar wavelet transform of the elements,
Figure GDA0002979553400000046
is a k-1 layer 2n-1The Haar wavelet transform of the elements has the iteration initial value:
Figure GDA0002979553400000047
suppose A2nIs a 2n×2nCalculating the tensor product of the unitary matrix of
Figure GDA0002979553400000048
And
Figure GDA0002979553400000049
can obtain
Figure GDA00029795534000000410
Figure GDA00029795534000000411
According to the formulas (1), (2), (3) and (4), when k is n-1, the complexity is designed to be theta (n)2) Wherein k and n are positive integers.
In the foregoing solution, it is preferable that the quantum circuit design implementation process of the first multi-layer quantum Haar wavelet inverse transform is as follows: and (3) performing inverse operation on the formula (2) to obtain a first multilayer quantum Haar wavelet inverse transformation iterative formula as follows:
Figure GDA00029795534000000412
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA00029795534000000413
is the sign of the operation of the tensor product,
Figure GDA00029795534000000414
is I2The product of the n tensors of (a),
Figure GDA00029795534000000415
is a matrix of uniform shuffling permutations,
Figure GDA00029795534000000416
is k layer 2nThe Haar wavelet inverse transform of the elements,
Figure GDA00029795534000000417
is a k-1 layer 2n-1And performing Haar wavelet inverse transformation on the elements, wherein the iteration initial value is as follows:
Figure GDA00029795534000000418
according to the formula (5), when k is n-1, the complexity is designed to be theta (n)2) Wherein k and n are positive integers.
In the foregoing solution, it is preferable that the design implementation process of the quantum wire of the second multilayer quantum Haar wavelet transform is as follows:
suppose that
Figure GDA0002979553400000051
Is k layer 2nA Haar wavelet transform of the elements,
Figure GDA0002979553400000052
is a k-1 layer 2n-1Elemental Haar wavelet transform, then the classical multi-layer quantum Haar transform can be defined as:
Figure GDA0002979553400000053
wherein
Figure GDA0002979553400000054
Is a uniform shuffling permutation matrix, H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000055
is I2The product of the n tensors. According to the principle of the expanded tensor product operation,
Figure GDA0002979553400000056
convertible to equivalent tensor products
Figure GDA0002979553400000057
Will be provided with
Figure GDA0002979553400000058
Substituting into formula (6), providedThe iterative formula for calculating the second multilayer quantum Haar transform is as follows:
Figure GDA0002979553400000059
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA00029795534000000510
is the sign of the operation of the tensor product,
Figure GDA00029795534000000511
is I2The product of the n tensors of (a),
Figure GDA00029795534000000512
is a matrix of uniform shuffling permutations,
Figure GDA00029795534000000513
is k layer 2nA Haar wavelet transform of the elements,
Figure GDA00029795534000000514
is a k-1 layer 2n-1The Haar wavelet transform of the elements has the iteration initial value:
Figure GDA00029795534000000515
when k is n-1, the complexity is designed to be theta (n)2) The quantum circuit of the first multilayer quantum Haar wavelet transform, wherein k and n are both positive integers.
In the above scheme, it is preferable that the design implementation process of the quantum wire of the second multi-layer quantum Haar wavelet inverse transform is as follows: and (3) performing inverse operation on the formula (7) to obtain a second multilayer quantum Haar wavelet inverse transformation iterative formula as follows:
Figure GDA00029795534000000516
whereinH and I2Is a single qubit gate in figure 1,
Figure GDA00029795534000000517
is the sign of the operation of the tensor product,
Figure GDA00029795534000000518
is I2Tensor product of n times, P2,2n-1Is a matrix of uniform shuffling permutations,
Figure GDA00029795534000000519
is k layer 2nThe Haar wavelet inverse transform of the elements,
Figure GDA0002979553400000061
is a k-1 layer 2n-1And performing Haar wavelet inverse transformation on the elements, wherein the iteration initial value is as follows:
Figure GDA0002979553400000062
according to the formula (8), when k is n-1, the complexity is designed to be theta (n)2) Wherein k and n are positive integers.
The invention has the advantages and effects that:
1. compared with the prior art for realizing the quantum Haar wavelet transform, the invention designs 2 realization lines of multilayer quantum Haar wavelet transform and 2 realization lines of multilayer quantum Haar wavelet inverse transform, thereby constructing a relatively complete quantum Haar wavelet transform system. The invention is the improvement and improvement of the existing quantum Haar wavelet transform realization technology.
2. Compared with the classical Haar wavelet transform realization technology, the quantum Haar wavelet transform realized by using the quantum circuit is an efficient transformation method, and the complexity of the realization circuit of the quantum Haar transform designed by the invention is theta (n)2) And the implementation complexity of the classical fast Haar wavelet transform is theta (2)n)。
3. The invention is suitable for a plurality of practical information processing application fields, for example, algorithms such as image compression, denoising, encryption and decryption need high-efficiency Haar wavelet transformation, and has great significance for quantum computing theory perfection and application popularization.
Drawings
FIG. 1 is a representation of a basic quantum gate and corresponding matrix of the present invention;
FIG. 2 shows the present invention
Figure GDA0002979553400000063
The quantum implementation circuit diagram of (1);
FIG. 3 shows the present invention
Figure GDA0002979553400000064
The quantum implementation circuit diagram of (1);
FIG. 4 is a graph of the product of the expansion tensor of the present invention
Figure GDA0002979553400000065
And
Figure GDA0002979553400000066
the quantum implementation circuit diagram of (1);
FIG. 5 is a circuit diagram for implementing the first K (1. ltoreq. K < n-1) layer quantum Haar wavelet transform according to the present invention;
FIG. 6 is a circuit diagram of the implementation of the first n-1-layer quantum Haar wavelet transform of the present invention;
FIG. 7 is a circuit diagram for implementing inverse wavelet transform of quantum Haar of the first K (K is more than or equal to 1 and less than n-1) layer according to the present invention;
FIG. 8 is a circuit diagram of the implementation of the first n-1-layer quantum Haar inverse wavelet transform according to the present invention;
FIG. 9 is a circuit diagram of the second K (1. ltoreq. K < n-1) quantum Haar wavelet transform implementation of the present invention;
FIG. 10 is a circuit diagram of the implementation of the second n-1-layer quantum Haar wavelet transform of the present invention;
FIG. 11 is a circuit diagram of the second K (1. ltoreq. K < n-1) quantum Haar inverse wavelet transform implementation of the present invention;
FIG. 12 is a circuit diagram of the implementation of the second n-1-layer quantum Haar inverse wavelet transform according to the present invention;
FIG. 13 is a circuit diagram of the implementation of the first k-1-layer Haar wavelet transform of the present invention;
FIG. 14 is a circuit diagram of the implementation of the first n-1-2-layer Haar wavelet transform of the present invention;
fig. 15 is a circuit diagram of the first implementation of the k-1-layer Haar inverse wavelet transform according to the present invention;
fig. 16 is a circuit diagram of the implementation of the first n-1-2-layer Haar inverse wavelet transform of the present invention;
FIG. 17 is a diagram of a second implementation of the second k-1-level Haar wavelet transform of the present invention;
fig. 18 is a circuit diagram of a second n-1-2-layer Haar wavelet transform implementation of the present invention;
fig. 19 is a circuit diagram of the second k-1-layer Haar inverse wavelet transform implementation of the present invention;
fig. 20 is a circuit diagram of the implementation of the second n-1-2-layer Haar inverse wavelet transform according to the present invention.
Detailed Description
The present invention is further illustrated by the following examples.
Example 1:
a method for realizing quantum circuit design by quantum Haar wavelet transform is disclosed, wherein quantum computation is combined with classical Haar wavelet transform technology to obtain quantum Haar wavelet transform; the quantum Haar wavelet transform is designed into a quantum circuit of the first multilayer quantum Haar wavelet transform according to the expanded tensor product operation principle.
Is provided with
Figure GDA0002979553400000081
Is k layer 2nA Haar wavelet transform of the elements,
Figure GDA0002979553400000082
is a k-1 layer 2n-1Elemental Haar wavelet transform, then the classical multi-layer quantum Haar transform can be defined as:
Figure GDA0002979553400000083
wherein
Figure GDA0002979553400000084
Is a uniform shuffling permutation matrix, H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000085
is I2The product of the n tensors.
According to the principle of the expanded tensor product operation, obtaining
Figure GDA0002979553400000086
Substituting the formula into the formula (1), and designing a first iteration formula of multilayer quantum Haar transformation as follows:
Figure GDA0002979553400000087
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000088
is the sign of the operation of the tensor product,
Figure GDA0002979553400000089
is I2The product of the n tensors of (a),
Figure GDA00029795534000000810
is a matrix of uniform shuffling permutations,
Figure GDA00029795534000000811
is k layer 2nA Haar wavelet transform of the elements,
Figure GDA00029795534000000812
is a k-1 layer 2n-1The Haar wavelet transform of the elements has the iteration initial value:
Figure GDA00029795534000000813
suppose A2nIs a 2n×2nCalculating the tensor product of the unitary matrix of
Figure GDA00029795534000000814
And
Figure GDA00029795534000000815
can obtain
Figure GDA00029795534000000816
Figure GDA00029795534000000817
The quantum wires for these two tensor products are shown in figure 4.
Combining the formulas (1), (2), (3) and (4), when K is equal to n-1, the quantum implementation circuit of the first multilayer quantum Haar wavelet transform is shown in fig. 5, and when K is equal to or less than 1 and less than n-1, the quantum implementation circuit of the first multilayer quantum Haar wavelet transform is shown in fig. 6. Since the process in the conversion process of the formulas (1), (2), (3), and (4) is mainly an iterative process and is common knowledge of those skilled in the art, the conversion process will not be described in detail.
The complexity of a quantum wire refers to the total number of single and double quantum bit gates that build the quantum wire. As shown in FIGS. 5 and 6, the complexity of the quantum implementation circuit of the first multi-layer quantum Haar wavelet transform is theta (n)2)。
First 1-layer quantum Haar wavelet transform designed by the invention
Figure GDA0002979553400000091
The implementation route of (2) is shown in fig. 13. Substituting n-3 and k-1 into formula (1) to obtain
Figure GDA0002979553400000092
The quantum wire in fig. 13 is obtained by implementing equation (9).
First 2-layer quantum Haar wavelet transform designed by the invention
Figure GDA0002979553400000093
The implementation route of (2) is shown in fig. 14. Substituting n-3 and k-n-1-2 into formula (1) yields
Figure GDA0002979553400000094
The quantum wire in fig. 14 is obtained by implementing equation (10).
Example 2:
a method for realizing quantum circuit design by quantum Haar wavelet transform is disclosed, wherein quantum computation is combined with classical Haar wavelet transform technology to obtain quantum Haar wavelet transform; and (3) designing quantum lines of the first multilayer quantum Haar wavelet inverse transformation according to the expanded tensor product operation principle by the quantum Haar wavelet transformation.
And (3) performing inverse operation on the formula (2) to obtain a first multilayer quantum Haar wavelet inverse transformation iterative formula as follows:
Figure GDA0002979553400000095
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000096
is the sign of the operation of the tensor product,
Figure GDA0002979553400000097
is I2The product of the n tensors of (a),
Figure GDA0002979553400000098
is a uniform shuffle permutation matrix, the corresponding quantum wires are shown in figure 3.
Figure GDA0002979553400000099
Is k layer 2nThe Haar wavelet inverse transform of the elements,
Figure GDA00029795534000000910
is a k-1 layer 2n-1And performing Haar wavelet inverse transformation on the elements, wherein the iteration initial value is as follows:
Figure GDA0002979553400000101
according to the formula (5), when K is equal to n-1, the quantum implementation circuit of the first multi-layer quantum Haar wavelet inverse transformation is shown in fig. 7, and when K is equal to or less than 1 and less than n-1, the quantum implementation circuit of the first multi-layer quantum Haar wavelet inverse transformation is shown in fig. 8.
As shown in FIGS. 7 and 8, the complexity of the quantum implementation circuit of the first multilayer quantum Haar inverse wavelet is θ (n)2)。
First layer 1 quantum Haar wavelet inverse transform designed by the invention
Figure GDA0002979553400000102
The implementation route of (2) is shown in fig. 15. Substituting n-3 and k-1 into equation (4) yields
Figure GDA0002979553400000103
The quantum wire in fig. 15 is obtained by implementing equation (11).
First 2-layer quantum Haar wavelet inverse transform designed by the invention
Figure GDA0002979553400000104
The implementation route of (2) is shown in fig. 16. Substituting n-3 and k-n-1-2 into equation (4) yields
Figure GDA0002979553400000105
The quantum wire in fig. 16 is obtained by implementing equation (12).
Example 3:
a method for realizing quantum circuit design by quantum Haar wavelet transform is disclosed, wherein quantum computation is combined with classical Haar wavelet transform technology to obtain quantum Haar wavelet transform; and (3) designing a quantum line of the second multilayer quantum Haar wavelet transform according to the expanded tensor product operation principle by the quantum Haar wavelet transform.
Suppose that
Figure GDA0002979553400000106
Is k layer 2nA Haar wavelet transform of the elements,
Figure GDA0002979553400000107
is a k-1 layer 2n-1Elemental Haar wavelet transform, then the classical multi-layer quantum Haar transform can be defined as:
Figure GDA0002979553400000108
wherein
Figure GDA0002979553400000111
Is a uniform shuffling permutation matrix, H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000112
is I2The product of the n tensors. According to the principle of the expanded tensor product operation,
Figure GDA0002979553400000113
convertible to equivalent tensor products
Figure GDA0002979553400000114
Will be provided with
Figure GDA0002979553400000115
Substituting the formula (6), and designing an iterative formula of a second multilayer quantum Haar transform as follows:
Figure GDA0002979553400000116
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000117
is the sign of the operation of the tensor product,
Figure GDA0002979553400000118
is I2The product of the n tensors of (a),
Figure GDA0002979553400000119
is a matrix of uniform shuffling permutations,
Figure GDA00029795534000001110
is k layer 2nA Haar wavelet transform of the elements,
Figure GDA00029795534000001111
is a k-1 layer 2n-1The Haar wavelet transform of the elements has the iteration initial value:
Figure GDA00029795534000001112
when K is equal to n-1, the quantum implementation circuit of the second multilayer quantum Haar wavelet transform is shown in FIG. 9, and when K is equal to or less than 1 and less than n-1, the quantum implementation circuit of the second multilayer quantum Haar wavelet transform is shown in FIG. 10.
As shown in FIGS. 9 and 10, the complexity of the quantum implementation circuit of the second multi-layer quantum Haar wavelet transform is θ (n)2)。
Second 1-layer quantum Haar wavelet transform designed by the invention
Figure GDA00029795534000001113
The implementation route of (2) is shown in fig. 17. Substituting n-3 and k-1 into equation (5) yields
Figure GDA00029795534000001114
The quantum wire in fig. 17 is obtained by implementing equation (13).
Second 2-layer quantum Haar wavelet transform designed by the invention
Figure GDA00029795534000001115
The implementation route of (2) is shown in fig. 18. Substituting n-3 and k-n-1-2 into equation (5) yields
Figure GDA00029795534000001116
The quantum wire in fig. 18 is obtained by implementing equation (14).
Example 4:
a method for realizing quantum circuit design by quantum Haar wavelet transform is disclosed, wherein quantum computation is combined with classical Haar wavelet transform technology to obtain quantum Haar wavelet transform; and (3) designing a quantum circuit of a second multilayer quantum Haar wavelet inverse transformation according to the expanded tensor product operation principle by the quantum Haar wavelet transformation.
And (3) performing inverse operation on the formula (7) to obtain a second multilayer quantum Haar wavelet inverse transformation iterative formula as follows:
Figure GDA0002979553400000121
wherein H and I2Is a single qubit gate in figure 1,
Figure GDA0002979553400000122
is the sign of the operation of the tensor product,
Figure GDA0002979553400000123
is I2The product of the n tensors of (a),
Figure GDA0002979553400000124
is a uniform shuffle permutation matrix, the corresponding quantum wires are shown in figure 3.
Figure GDA0002979553400000125
Is k layer 2nThe Haar wavelet inverse transform of the elements,
Figure GDA0002979553400000126
is a k-1 layer 2n-1And performing Haar wavelet inverse transformation on the elements, wherein the iteration initial value is as follows:
Figure GDA0002979553400000127
when K is equal to n-1, the quantum implementation circuit of the second multi-layer quantum Haar wavelet inverse transformation is shown in fig. 11, and when K is equal to or less than 1 and less than n-1, the quantum implementation circuit of the second multi-layer quantum Haar wavelet inverse transformation is shown in fig. 12.
From fig. 11 and 12, the complexity of the quantum implementation circuit of the second multi-layer quantum Haar wavelet inverse transform is Θ (n)2)。
Second layer 1 quantum Haar wavelet inverse transform designed by the invention
Figure GDA0002979553400000128
The implementation route of (2) is shown in fig. 19. Substituting n-3 and k-1 into equation (6) yields
Figure GDA0002979553400000129
The quantum wire in fig. 19 is obtained by implementing equation (15).
Second 2-layer quantum Haar wavelet inverse transform designed by the invention
Figure GDA0002979553400000131
The implementation route of (2) is shown in fig. 20. Substituting n-3 and k-n-1-2 into equation (6) yields
Figure GDA0002979553400000132
The quantum wire in fig. 20 is obtained by implementing equation (16).
While the preferred embodiments of the present invention have been described in detail, it is to be understood that the invention is not limited thereto, and that various equivalent modifications and substitutions may be made by those skilled in the art without departing from the spirit of the present invention and are intended to be included within the scope of the present application.

Claims (4)

1. A method for realizing quantum circuit design by quantum Haar wavelet transform is characterized in that: the method combines quantum computation and a classical Haar wavelet transform technology to obtain quantum Haar wavelet transform; designing quantum circuits of 2 quantum Haar wavelet transforms and 2 quantum Haar wavelet inverse transforms according to the expanded tensor product operation principle by using the quantum Haar wavelet transforms;
2 quantum lines of the quantum Haar wavelet transform are respectively a quantum line of the first multilayer quantum Haar wavelet transform and a quantum line of the second multilayer quantum Haar wavelet transform;
2 quantum circuits of inverse quantum Haar wavelet transform are quantum circuits of first multilayer quantum Haar wavelet inverse transform and quantum circuits of second multilayer quantum Haar wavelet inverse transform respectively;
the quantum circuit design realization process of the first multilayer quantum Haar wavelet transform comprises the following steps: is provided with
Figure FDA0002876097530000011
Is k layer 2nA Haar wavelet transform of the elements,
Figure FDA0002876097530000012
is a k-1 layer 2n-1Elemental Haar wavelet transform, then the classical multi-layer quantum Haar transform can be defined as:
Figure FDA0002876097530000013
wherein
Figure FDA0002876097530000014
Is a uniform shuffling permutation matrix, H and I2Is a single-quantum-bit gate that,
Figure FDA0002876097530000015
is I2The product of the n tensors of (a),
Figure FDA0002876097530000016
is a tensor product operation symbol;
according to the principle of the expanded tensor product operation, obtaining
Figure FDA0002876097530000017
Substituting the formula into the formula (1), and designing a first iteration formula of multilayer quantum Haar transformation as follows:
Figure FDA0002876097530000018
wherein H and I2Is a single-quantum-bit gate that,
Figure FDA0002876097530000019
is I2The product of the n tensors of (a),
Figure FDA00028760975300000110
is a matrix of uniform shuffling permutations,
Figure FDA00028760975300000111
is k layer 2nA Haar wavelet transform of the elements,
Figure FDA00028760975300000112
is a k-1 layer 2n-1The Haar wavelet transform of the elements has the iteration initial value:
Figure FDA00028760975300000113
suppose that
Figure FDA00028760975300000114
Is a 2n×2nCalculating the tensor product of the unitary matrix of
Figure FDA00028760975300000115
And
Figure FDA00028760975300000116
can obtain
Figure FDA0002876097530000021
Figure FDA0002876097530000022
Wherein the content of the first and second substances,
Figure FDA0002876097530000023
is a 2n×2nUnitary matrix of
According to the formulas (1), (2), (3) and (4), when k is n-1, the complexity is designed to be theta (n)2) Wherein k and n are positive integers.
2. The method for quantum Haar wavelet transform to realize quantum wire design according to claim 1, wherein: the quantum circuit design realization process of the first multilayer quantum Haar wavelet inverse transformation comprises the following steps: and (3) performing inverse operation on the formula (2) to obtain a first multilayer quantum Haar wavelet inverse transformation iterative formula as follows:
Figure FDA0002876097530000024
wherein H and I2Is a single-quantum-bit gate that,
Figure FDA0002876097530000025
is the sign of the operation of the tensor product,
Figure FDA0002876097530000026
is I2The product of the n tensors of (a),
Figure FDA0002876097530000027
is an inverse of the uniform shuffling permutation matrix,
Figure FDA0002876097530000028
is k layer 2nThe Haar wavelet inverse transform of the elements,
Figure FDA0002876097530000029
is a k-1 layer 2n-1And performing Haar wavelet inverse transformation on the elements, wherein the iteration initial value is as follows:
Figure FDA00028760975300000210
according to the formula (5), when k is n-1, the complexity is designed to be theta (n)2) Wherein k and n are positive integers.
3. The method for quantum Haar wavelet transform to realize quantum wire design according to claim 1, wherein: the quantum circuit design realization process of the second multilayer quantum Haar wavelet transform comprises the following steps: suppose that
Figure FDA00028760975300000211
Is k layer 2nA Haar wavelet transform of the elements,
Figure FDA00028760975300000212
is a k-1 layer 2n-1Elemental Haar wavelet transform, then the classical multi-layer quantum Haar transform can be defined as:
Figure FDA00028760975300000213
wherein
Figure FDA0002876097530000031
Is a matrix of uniform shuffling permutations,
Figure FDA0002876097530000032
is the sign of the tensor product operation, H and I2Is a single-quantum-bit gate that,
Figure FDA0002876097530000033
is I2The product of n tensors; according to the principle of the expanded tensor product operation,
Figure FDA0002876097530000034
convertible to equivalent tensor products
Figure FDA0002876097530000035
Will be provided with
Figure FDA0002876097530000036
Substituting the formula (6), and designing an iterative formula of a second multilayer quantum Haar transform as follows:
Figure FDA0002876097530000037
wherein H and I2Is a single-quantum-bit gate that,
Figure FDA0002876097530000038
is I2The product of the n tensors of (a),
Figure FDA0002876097530000039
is a matrix of uniform shuffling permutations,
Figure FDA00028760975300000310
is k layer 2nA Haar wavelet transform of the elements,
Figure FDA00028760975300000311
is a k-1 layer 2n-1The Haar wavelet transform of the elements has the iteration initial value:
Figure FDA00028760975300000312
when k is n-1, the complexity is designed to be theta (n)2) The quantum circuit of the first multilayer quantum Haar wavelet transform, wherein k and n are both positive integers.
4. The method for quantum Haar wavelet transform to realize quantum wire design according to claim 3, wherein: the quantum circuit design realization process of the second multilayer quantum Haar wavelet inverse transformation comprises the following steps: and (3) performing inverse operation on the formula (7) to obtain a second multilayer quantum Haar wavelet inverse transformation iterative formula as follows:
Figure FDA00028760975300000313
wherein H and I2Is a single-quantum-bit gate that,
Figure FDA00028760975300000314
is the sign of the operation of the tensor product,
Figure FDA00028760975300000315
is I2The product of the n tensors of (a),
Figure FDA00028760975300000316
is contrary toThe permutation matrix is evenly shuffled and the matrix is,
Figure FDA00028760975300000317
is k layer 2nThe Haar wavelet inverse transform of the elements,
Figure FDA00028760975300000318
is a k-1 layer 2n-1And performing Haar wavelet inverse transformation on the elements, wherein the iteration initial value is as follows:
Figure FDA00028760975300000319
according to the formula (8), when k is n-1, the complexity is designed to be theta (n)2) Wherein k and n are positive integers.
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CN108154240B (en) * 2017-12-29 2020-07-07 合肥本源量子计算科技有限责任公司 Low-complexity quantum line simulation system
CN108363677B (en) * 2018-01-15 2024-04-26 湖南哥禄安科技有限公司 Method for realizing quantum circuit design by two-dimensional quantum Haar wavelet packet transformation and inverse transformation
CN108255784B (en) * 2018-01-15 2024-01-09 宁波亚翔电子科技有限公司 Multilayer quantum D (4) Method for realizing quantum circuit design by wavelet packet transformation and inverse transformation
CN108388948B (en) * 2018-03-13 2021-11-23 广西师范大学 Design method for type conversion from quantum image to quantum real signal
CN108710951A (en) * 2018-05-17 2018-10-26 合肥本源量子计算科技有限责任公司 A kind of method and system of structure quantum wire
CN108964887B (en) * 2018-05-26 2021-04-20 广西师范大学 Quantum image encryption method based on NASS quantum image representation
CN108898228B (en) * 2018-06-21 2024-03-08 广西师范大学 Quantum adder design method without damaging source operands
CN108847050B (en) * 2018-08-10 2020-10-30 湖南小闸科技有限公司 Data processing system and method for preventing high concurrency by sharing parking space
TWI802206B (en) * 2022-01-04 2023-05-11 國立成功大學 Quantum circuit for daubechies' d6 wavelet transform and inverse transform and manufacturing method thereof

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101216939A (en) * 2008-01-04 2008-07-09 江南大学 A multi-resolution medical image registration method based on quantum behaviors particle swarm algorithm
EP2618269A1 (en) * 2012-01-17 2013-07-24 Siemens Aktiengesellschaft A method for processing of measurements from several sensors
CN103971328A (en) * 2014-05-05 2014-08-06 华东交通大学 Storage, design and implementation method for multi-dimensional quantum gray image and multi-dimensional quantum colorful image
CN104285175A (en) * 2012-05-02 2015-01-14 法国国家科学研究中心 Method and apparatus for single-particle localization using wavelet analysis

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20140327737A1 (en) * 2013-05-01 2014-11-06 Raymond John Westwater Method and Apparatus to Perform Optimal Visually-Weighed Quantization of Time-Varying Visual Sequences in Transform Space

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101216939A (en) * 2008-01-04 2008-07-09 江南大学 A multi-resolution medical image registration method based on quantum behaviors particle swarm algorithm
EP2618269A1 (en) * 2012-01-17 2013-07-24 Siemens Aktiengesellschaft A method for processing of measurements from several sensors
CN104285175A (en) * 2012-05-02 2015-01-14 法国国家科学研究中心 Method and apparatus for single-particle localization using wavelet analysis
CN103971328A (en) * 2014-05-05 2014-08-06 华东交通大学 Storage, design and implementation method for multi-dimensional quantum gray image and multi-dimensional quantum colorful image

Non-Patent Citations (5)

* Cited by examiner, † Cited by third party
Title
Efficient Quantum Transforms;Peter Hoyer;《arXiv preprint quant-ph/9702028》;19970211;1-30 *
Imperfection efects for multiple applications of the quantum wavelet transform;M.Terraneo et al.;《arXiv preprint quant-ph/0303043》;20030309;1-11 *
Wavelets and wavelet packets on quantum computers;A. Klappenecker;《arXiv preprint quant-ph》;19990903;1-11 *
量子小波变换算法及线路实现;朱轩溢，等;《计算机应用与软件》;20080930;第25卷(第9期);239-241,247 *
量子小波变换算法设计与应用研究;张才智;《中国优秀硕士学位论文全文数据库 信息科技辑》;20090315(第03期);I136-56 *

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