CN107180013A - A kind of method that quantum D (4) wavelet transformation realizes quantum wire design - Google Patents
A kind of method that quantum D (4) wavelet transformation realizes quantum wire design Download PDFInfo
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Abstract
The present invention provides a kind of method that quantum D (4) wavelet transformation realizes quantum wire design, belongs to quantum information process field, present invention design individual layer quantum D(4)Wavelet transformation, it is to existing quantum D to have used two spin matrixs to replace in general unitary matrice, method(4)A kind of innovation of wavelet transformation technique.From quantum D(4)Wavelet transformation and quantum D(4)Wavelet inverse transformation realize network complexity analysis understand, for one 2nThe data set of individual element, quantum D(4)Wavelet transformation and quantum D(4)The complexity of the circuit of wavelet inverse transformation is all Θ (n2), this is other classical quick D(4)What wavelet transformation was unable to reach.The present invention is applied to many actual information processing application fields, for example, the compression of image, denoising, encryption and decryption scheduling algorithm are required for efficient D(4)Wavelet transformation, and perfect and application the popularization of quantum calculation theory is of great importance.
Description
Technical field
Specifically it is to be related to a kind of quantum D (4) wavelet transformation realization amount the present invention relates to quantum information process field
The method of sub-line road design.
Background technology
Quantum calculation is the product that quantum mechanics and computer science are combined, the concurrency of quantum calculation, additivity and
Its uncertainty measured is that quantum computer is basic better than classic computer.In face of such advantage, the research of quantum information
Just seem necessary, it turns into countries in the world Strategic Competition focus, and national " 13 " planning outline is by quantum communications with measuring
Sub- computer is classified as the major scientific and technological projects (scientific and technical innovation 2030- major projects) of national strategy intention, and gives priority to quantum
Information etc. leads the subversiveness technology of industry transformation.
Many western wavelet transformations of shellfish (Daubechies wavelet transformation) are that the most-often used small echo arrived turns
Alternatively one, it is also a kind of orthogonal wavelet.D(4)Wavelet transformation is a kind of fairly simple conversion of the western small echo series of transformations of many shellfishes, very
Easily change and realize via Fast Wavelet, therefore have in field of information processing important application.Corresponding quantum D(4)Conversion is
The important tool algorithm of quantum information processing, the extensive application in Image Coding, rim detection, image watermark scheduling algorithm.
In traditional counting, information unit represents that it only has two states with bit (Bit)：0 state or 1 state.In quantum meter
In calculation, information unit represents that it has two basic quantum states with quantum bit (Qubit) | 0>With | 1>, basic quantum state is referred to as
For ground state (Basis State).One quantum bit can be the linear combination of two ground state, be commonly referred to as superposition state
(Superposition), it is represented by | ψ>=a | 0>+b|1>.Wherein a and b are two plural numbers, are met | a |2+|b|2=1, because
This is also referred to as probability amplitude.Ground state | 0>With | 1>, availability vector is expressed as：
Their dual vector is represented by<0 |=[1 0],<1 |=[0 1].
Tensor product (tensorproduct) is to be combined small vector space, constitutes one kind of bigger vector space
Method, uses symbolRepresent, it has following implication：
Assuming that it is two complex matrix of m × m that U, which is n × n and V,
So
Assuming that two unitary matrice collection are combined into：WithIn have m n × n matrix, In have n m × m matrix.The tensor product of extension is mn × mn matrixWherein
WhenIn each matrix it is identical, Ai=A, nowIt can be write asIf simultaneouslyIn it is every
The all identical B of individual matrixi=B, the tensor product at this moment extendedIt is degenerated to common tensor product
Quantum wire can be made up of the quantum bit door of a sequence, in the expression figure of quantum wire, every line all tables
Show the line of quantum wire, the execution sequence of quantum wire is from left to right.Quantum bit door can be conveniently with matrix form
Represent, single quantum bit door can be represented with the unitary matrice U of one 2 × 2, i.e. U+U=I, wherein U+It is U associate matrix,
I is unit matrix.In double quantum bits door, most important is controlled not-gate, and it has the bit input and output of two quantum, respectively
It is control quantum bit and target quantum bit.When control bit is 1, represented with stain, when control bit is 0, use white point table
Show.Title, symbol and the corresponding matrix of some basic quantum bit gates represent to see Fig. 1.
Due to existing classical classical D(4)Wavelet transformation realizes that the complexity of electronic circuitry design is Θ (n2n), than
It is more complicated, the demand of society could not be met very well.Therefore need to design the lower side for realizing electronic circuitry design of complexity
Method.
The content of the invention
The present invention provides a kind of quantum D(4)The method that wavelet transformation realizes quantum wire design, solves existing classical D(4)
Wavelet transformation realizes the problem of complexity of electronic circuitry design is high
The present invention solves the above problems by the following technical programs：
The special performance of the abundant quantum calculation such as quantum parallelism and quantum superposition, using the tensor product of extension, sets up D(4)The iterative formula of wavelet transformation, and the tensor product extended is realized using the controlled door of quantum, so as to realize quantum D(4)Small echo becomes
Change.Specifically, 2 quantum D are exactly designed according to the tensor product principle of operation of extension(4)Wavelet transformation and 2 quantum D(4)
Wavelet inverse transformation realizes circuit；
2 quantum D(4)Wavelet transformation realize circuit respectively include individual layer quantum D(4)Wavelet transformation realizes circuit and K+
1 layer of quantum D(4)Wavelet transformation realizes circuit；
2 quantum D(4)Wavelet inverse transformation realize circuit respectively include individual layer quantum D(4)Wavelet inverse transformation realizes circuit
With K+1 layers of quantum D(4)Wavelet inverse transformation realizes circuit.
In such scheme, preferably described individual layer quantum D(4)The detailed process for realizing circuit of wavelet transformation is：
D(4)Small echo nuclear matrix is defined as：
Wherein
Can be with tensor product representation
WhereinIt is tensor product oeprator,It is I2N tensor product,With
WithIt is two spin matrixs,
Unitary matriceIterative formula it is as follows：
Wherein X and I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N times
Amount product, iteration initial value is Q2=X；
Assuming thatIt is one 2n×2nUnitary matrice, calculate tensor productWithIt is available
By formula (3), it can obtainIt is substituted into formula
(2), so as to obtain individual layer quantum D(4)Wavelet transformation:
Using formula (5), complexity is designed for Θ (n2) individual layer quantum D(4)Wavelet transformation realizes circuit.
In such scheme, preferably described individual layer quantum D(4)The design process for realizing circuit of wavelet inverse transformation is：
Formula (5) is inverted, D is can obtain(4)Wavelet Kernel matrix inverse transformation is iterative and unitary matrice Q2nIt is iterative：
Wherein iteration initial value is (Q2)-1=X；
With reference to formula (6), complexity is designed for Θ (n2) individual layer quantum D(4)Wavelet inverse transformation realizes circuit.
In such scheme, preferably described K+1 layers of quantum D(4)Wavelet transformation realize circuit design implementation process be：
By k+1 layers of quantum D(4)Wavelet transformation is defined asUsing the tensor product of extension, it can obtain：
Wherein I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor
Product,For k layers of quantum D(4)Wavelet transformation,For individual layer quantum D (4) wavelet transformation, iteration initial value is1
≤ k≤n-2, k, n are positive integer；
With reference to formula (5) and (7), complexity is designed for Θ (n2) K+1 layer quantum D(4)Wavelet transformation realizes line
Road.
In such scheme, preferably described K+1 layers of quantum D(4)Wavelet inverse transformation realize circuit realize detailed process
For：
OrderFor k+1 layers of quantum D(4)Wavelet inverse transformation, inverts to formula (7), can obtain：
Wherein I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor
Product,For k layers of quantum D(4)Wavelet inverse transformation,For individual layer quantum D (4) wavelet inverse transformation (see formula (5)), change
It is for initial value1≤k≤n-2, k, n are positive integer；
With reference to formula (6) and (8), complexity is designed for Θ (n2) k+1 layer quantum D(4)Wavelet inverse transformation realizes line
Road.
Advantages of the present invention is with effect：
1st, of the invention and existing quantum D(4)Wavelet transformation realizes that technology is compared, present invention design individual layer quantum D(4)Small echo
Conversion, it is to existing quantum D to have used two spin matrixs to replace in general unitary matrice, method(4)Wavelet transformation technique
One kind innovation.
2nd, of the invention and existing quantum D(4)Wavelet transformation realizes that technology is compared, and the present invention devises multi-layer quantum D(4)It is small
Wave conversion and multi-layer quantum D(4)Wavelet inverse transformation realizes circuit, so as to build a complete quantum D relatively(4)Wavelet transformation
System.And prior art only realizes individual layer quantum D(4)Wavelet transformation, the present invention is real to existing quantum D (4) wavelet transformation
The perfect and improvement of existing technology.
3rd, the present invention and classical D(4)Wavelet transformation realizes that technology is compared, the quantum D that the present invention is realized using quantum wire(4)Wavelet transformation is a kind of efficient transform method, the quantum D that the present invention is designed(4)Wavelet transformation realizes the network complexity all
It is Θ (n2), and classical quick D(4)The implementation complexity of wavelet transformation is Θ (2n)。
4th, the special performance of the present invention fully quantum calculation such as quantum parallelism and quantum superposition, using of extension
Amount product, realizes individual layer quantum D first(4)Wavelet transformation and individual layer quantum D(4)The iterative formula of wavelet inverse transformation, then sets up many
Layer quantum D(4)The iterative formula of conversion and corresponding quantum D(4)The iterative formula of wavelet inverse transformation.And using quantum wire come real
Existing quantum D(4)Wavelet transformation and corresponding quantum D(4)Wavelet inverse transformation.
Brief description of the drawings
Fig. 1 is the expression figure of fundamental quantity cervical orifice of uterus of the present invention and homography；
Fig. 2 is present invention extension tensor productLine map is realized with corresponding quantum；
Fig. 3 is individual layer quantum D of the present invention(4)Wavelet transformation realizes line map；
Fig. 4 is individual layer quantum D of the present invention(4)The simplification symbol table diagram for realizing circuit of wavelet transformation；
Fig. 5 is individual layer quantum D of the present invention(4)Wavelet inverse transformation realizes line map；
Fig. 6 is individual layer quantum D of the present invention(4)The simplification symbol table diagram for realizing circuit of wavelet inverse transformation；
Fig. 7 is k+1 layers of quantum D of the present invention(4)Wavelet transformation realizes line map；
Fig. 8 is k+1 layers of quantum D of the present invention(4)Wavelet inverse transformation realizes line map；
Fig. 9 is individual layer quantum D of the present invention(4)Wavelet transformation realizes line map；
Figure 10 is individual layer quantum D of the present invention(4)Wavelet inverse transformation realizes line map；
Figure 11 is two layers of quantum D of the present invention(4)Wavelet transformation realizes line map；
Figure 12 is two layers of quantum D of the present invention(4)Wavelet inverse transformation realizes line map.
Embodiment
The invention will be further described with reference to embodiments.
Embodiment 1：
A kind of method that quantum D (4) wavelet transformation realizes quantum wire design, by quantum calculation and classics D(4)Small echo becomes
The technology of changing, which is combined, obtains quantum D(4)Wavelet transformation.D(4)Wavelet transformation designs list according to the tensor product principle of operation of extension
Layer quantum D(4)Wavelet transformation realizes circuit circuit.
D(4)Small echo nuclear matrix is defined as：
Wherein
Can be with tensor product representation
WhereinIt is tensor product oeprator,It is I2N tensor product,With
WithIt is two spin matrixs,
Unitary matriceIterative formula it is as follows：
Wherein X and I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N times
Amount product, iteration initial value is Q2=X；
Assuming thatIt is one 2n×2nUnitary matrice, calculate tensor productWithIt is available
By formula (3), it can obtainIt is substituted into formula
(2), so as to obtain individual layer quantum D(4)Wavelet transformation:
The corresponding quantum wire of the two tensor products is as shown in Figure 2.With reference to formula ((5), individual layer quantum D(4)Wavelet transformation
Quantum realize circuit as shown in figure 3, its simplify symbol represent as shown in Figure 4.Known by Fig. 5, individual layer quantum D(4)Wavelet transformation
Quantum realizes that the complexity of circuit is Θ (n2)。
The individual layer quantum D that the present invention is designed(4)Wavelet transformationRealize that circuit is as shown in Figure 9.N=3 is substituted into formula
(5), obtain
Realize that formula (9) just obtains the quantum wire in Fig. 9.
Embodiment 2：
A kind of method that quantum D (4) wavelet transformation realizes quantum wire design, by quantum calculation and classics D(4)Small echo becomes
The technology of changing, which is combined, obtains quantum D(4)Wavelet transformation.D(4)Wavelet transformation designs list according to the tensor product principle of operation of extension
Layer quantum D(4)Wavelet inverse transformation realizes circuit circuit.
Formula (5) is inverted, D is can obtain(4)Wavelet Kernel matrix inverse transformation is iterative and unitary matrice Q2nIt is iterative：
Wherein iteration initial value is (Q2)-1=X；
With reference to formula (6), individual layer quantum D(4)The quantum of wavelet inverse transformation realizes circuit as shown in figure 5, it simplifies symbol table
Show as shown in Figure 6.Known by Fig. 5, individual layer quantum D(4)The quantum of wavelet inverse transformation realizes that the complexity of circuit is Θ (n2)。
The individual layer quantum D that the present invention is designed(4)Wavelet inverse transformationRealize that circuit is as shown in Figure 10.N=3 is substituted into
Formula (6), is obtained
Realize that formula (10) just obtains the quantum wire in Figure 10.
Embodiment 3：
A kind of method that quantum D (4) wavelet transformation realizes quantum wire design, by quantum calculation and classics D(4)Small echo becomes
The technology of changing, which is combined, obtains quantum D(4)Wavelet transformation.D(4)Wavelet transformation designs K+ according to the tensor product principle of operation of extension
1 layer of quantum D(4)Wavelet transformation realizes circuit.
By k+1 layers of quantum D(4)Wavelet transformation is defined asUsing the tensor product of extension, it can obtain：
Wherein I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor
Product,For k layers of quantum D(4)Wavelet transformation,For individual layer quantum D (4) wavelet transformation, iteration initial value is1
≤ k≤n-2, k, n are positive integer；
With reference to formula (5) and (7), on the basis of the quantum wire in realizing Fig. 4, k+1 layers of quantum D(4)Wavelet transformation
Quantum realizes circuit as shown in fig. 7, and understanding that quantum realizes that the complexity of circuit is Θ (n2)。
Two layers of quantum D that the present invention is designed(4)Wavelet transformationRealize that circuit is as shown in figure 11.By n=3 and k=1 generations
Enter formula (7), obtain
Realize that formula (11) just obtains the quantum wire in Figure 11.
Embodiment 4：
A kind of method that quantum D (4) wavelet transformation realizes quantum wire design, by quantum calculation and classics D(4)Small echo becomes
The technology of changing, which is combined, obtains quantum D(4)Wavelet transformation.D(4)Wavelet transformation designs K+ according to the tensor product principle of operation of extension
1 layer of quantum D(4)Wavelet inverse transformation realizes circuit.
OrderFor k+1 layers of quantum D(4)Wavelet inverse transformation, inverts to formula (7), can obtain：
Wherein I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor
Product,For k layers of quantum D(4)Wavelet inverse transformation,For individual layer quantum D (4) wavelet inverse transformation (see formula (5)), change
It is for initial value1≤k≤n-2, k, n are positive integer；
With reference to formula (6) and (8), on the basis of the quantum wire in realizing Fig. 6, k+1 layers of quantum D(4)Wavelet inverse transformation
Quantum realize circuit as shown in figure 8, and understanding that quantum realizes that the complexity of circuit is Θ (n2)。
Two layers of quantum D that the present invention is designed(4)Wavelet inverse transformationRealize that circuit is as shown in figure 12.By n=3 and k
=1 substitutes into formula (8), obtains
Realize that formula (12) just obtains the quantum wire in Figure 12.
The special performance of the fully quantum calculation such as quantum parallelism and quantum superposition of the invention, using the tensor of extension
Product, realizes individual layer quantum D first(4)Wavelet transformation and individual layer quantum D(4)The iterative formula of wavelet inverse transformation, then sets up multilayer
Quantum D(4)The iterative formula of conversion and corresponding quantum D(4)The iterative formula of wavelet inverse transformation.And realized using quantum wire
Quantum D(4)Wavelet transformation and corresponding quantum D(4)Wavelet inverse transformation.
The preferred embodiment to the invention is illustrated above, but the present invention is not limited to embodiment,
Those skilled in the art can also make a variety of equivalent modifications on the premise of without prejudice to the invention spirit
Or replace, these equivalent modifications or replacement are all contained in scope of the present application.
Claims (5)
1. a kind of method that quantum D (4) wavelet transformation realizes quantum wire design, it is characterised in that：Methods described is by quantum meter
Calculate and classics D(4)Wavelet transformation technique, which is combined, obtains quantum D(4)Wavelet transformation, D(4)Wavelet transformation is according to the tensor of extension
Product principle of operation designs 2 quantum D(4)Wavelet transformation and 2 quantum D(4)Wavelet inverse transformation realizes circuit；
2 quantum D(4)Wavelet transformation realize circuit respectively include individual layer quantum D(4)Wavelet transformation realizes circuit and K+1 layers
Quantum D(4)Wavelet transformation realizes circuit；
2 quantum D(4)Wavelet inverse transformation realize circuit respectively include individual layer quantum D(4)Wavelet inverse transformation realizes circuit and K+
1 layer of quantum D(4)Wavelet inverse transformation realizes circuit.
2. the method that a kind of quantum D (4) wavelet transformation according to claim 1 realizes quantum wire design, its feature exists
In the individual layer quantum D(4)The detailed process for realizing circuit of wavelet transformation is：The tensor product principle of operation of extension is so as to obtain
Go out, D(4)Small echo nuclear matrix is defined as：
Wherein
Can be with tensor product representation
WhereinIt is tensor product oeprator,It is I2N tensor product,With withIt is two spin matrixs,
Unitary matriceIterative formula it is as follows：
Wherein X and I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor product,
Iteration initial value is Q2=X；
Assuming thatIt is one 2n×2nUnitary matrice, calculate tensor productWithIt is available
By formula (3), it can obtainIt is substituted into formula (2), from
And obtain individual layer quantum D(4)Wavelet transformation:
Using formula (5), complexity is designed for Θ (n2) individual layer quantum D(4)Wavelet transformation realizes circuit.
3. the method that a kind of quantum D (4) wavelet transformation according to claim 2 realizes quantum wire design, its feature exists
In：The individual layer quantum D(4)The design process for realizing circuit of wavelet inverse transformation is：
Formula (5) is inverted, D is can obtain(4)Wavelet Kernel matrix inverse transformation is iterative and unitary matriceIt is iterative：
Wherein iteration initial value is (Q2)-1=X；
With reference to formula (6), complexity is designed for Θ (n2) individual layer quantum D(4)Wavelet inverse transformation realizes circuit.
4. the method that a kind of quantum D (4) wavelet transformation according to claim 3 realizes quantum wire design, its feature exists
In：The K+1 layers of quantum D(4)Wavelet transformation realize circuit design implementation process be：OrderFor k+1 layers of quantum D(4)Small echo becomes
Change, using the tensor product of extension, can obtain：
Wherein I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor product,
For k layers of quantum D(4)Wavelet transformation,For individual layer quantum D (4) wavelet transformation, iteration initial value is1≤k≤n-
2, k, n is positive integer；
With reference to formula (5) and (7), complexity is designed for Θ (n2) K+1 layer quantum D(4)Wavelet transformation realizes circuit.
5. the method that a kind of quantum D (4) wavelet transformation according to claim 4 realizes quantum wire design, its feature exists
In：The K+1 layers of quantum D(4)The detailed process of realizing for realizing circuit of wavelet inverse transformation is：
OrderFor k+1 layers of quantum D(4)Wavelet inverse transformation, inverts to formula (7), can obtain：
Wherein I2It is single quantum bit door in Fig. 1,It is tensor product oeprator,It is I2N tensor product,For k layers of quantum D(4)Wavelet inverse transformation,It is small for individual layer quantum D (4).
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| CN108255784A (en) * | 2018-01-15 | 2018-07-06 | 广西师范大学 | Multi-layer quantum D(4)The method that quantum wire design is realized in wavelet package transforms and inverse transformation |
| CN108363677A (en) * | 2018-01-15 | 2018-08-03 | 广西师范大学 | The method that quantum wire design is realized in two-dimentional quantum Haar wavelet package transforms and inverse transformation |
| 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 |
| CN108898228A (en) * | 2018-06-21 | 2018-11-27 | 广西师范大学 | A kind of quantum adder designs method for not destroying source operand |
| CN110826719A (en) * | 2019-10-14 | 2020-02-21 | 合肥本源量子计算科技有限责任公司 | Quantum program processing method and device, storage medium and electronic device |
| 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 |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN108255784A (en) * | 2018-01-15 | 2018-07-06 | 广西师范大学 | Multi-layer quantum D(4)The method that quantum wire design is realized in wavelet package transforms and inverse transformation |
| CN108363677A (en) * | 2018-01-15 | 2018-08-03 | 广西师范大学 | The method that quantum wire design is realized in two-dimentional quantum Haar wavelet package transforms 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 |
| 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 |
| 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 |
| CN108898228A (en) * | 2018-06-21 | 2018-11-27 | 广西师范大学 | A kind of quantum adder designs method for not destroying source operand |
| CN108898228B (en) * | 2018-06-21 | 2024-03-08 | 广西师范大学 | Quantum adder design method without damaging source operands |
| CN110826719A (en) * | 2019-10-14 | 2020-02-21 | 合肥本源量子计算科技有限责任公司 | Quantum program processing method and device, storage medium and electronic device |
| CN110826719B (en) * | 2019-10-14 | 2022-08-16 | 合肥本源量子计算科技有限责任公司 | Quantum program processing method and device, storage medium and electronic device |
| 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 |
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