CN107180013A - A kind of method that quantum D (4) wavelet transformation realizes quantum wire design - Google Patents

Abstract

The invention provides a method for realizing quantum circuit design through quantum D(4) wavelet transform, which belongs to the field of quantum information processing. The present invention designs single-layer quantum D ⁽⁴⁾ wavelet transform, and uses two rotation matrices to replace general unitary matrices. The method is an innovation to the existing quantum D ⁽⁴⁾ wavelet transform technology. From the circuit complexity analysis of quantum D ⁽⁴⁾ wavelet transform and quantum D ⁽⁴⁾ wavelet inverse transform, we can see that for a data set with 2 ⁿ elements, quantum D ⁽⁴⁾ wavelet transform and quantum D ⁽⁴⁾ wavelet inverse The complexities of the transformed circuits are all Θ(n ² ), which cannot be achieved by other classical fast D ⁽⁴⁾ wavelet transforms. The present invention is applicable to many practical information processing application fields. For example, algorithms such as image compression, denoising, encryption and decryption all require efficient D ⁽⁴⁾ wavelet transform, and have great significance for the improvement of quantum computing theory and the promotion of applications. .

Description

A method of quantum D(4) wavelet transform to realize quantum circuit design

technical field

The invention relates to the field of quantum information processing, in particular to a method for realizing quantum circuit design through quantum D(4) wavelet transform.

Background technique

Quantum computing is the product of the combination of quantum mechanics and computer science. The parallelism, superposition and measurement uncertainty of quantum computing are fundamental to the superiority of quantum computers over classical computers. In the face of such advantages, the research of quantum information is very necessary. It has become the focus of strategic competition among countries in the world. The national "Thirteenth Five-Year Plan" lists quantum communication and quantum computers as major scientific and technological projects of the national strategic intention (Science and Technology Innovation 2030 -Major projects), and focus on the development of disruptive technologies such as quantum information that lead industrial transformation.

Daubechies wavelet transformation (Daubechies wavelet transformation) is one of the most commonly used wavelet transformations, and it is also an orthogonal wavelet. D ⁽⁴⁾ Wavelet transform is a relatively simple transform of the Dobecy wavelet series transform, which can be easily realized by fast wavelet transform, so it has an important application in the field of information processing. The corresponding quantum D ⁽⁴⁾ transform is an important tool algorithm for quantum information processing, and it is widely used in algorithms such as image coding, edge detection, and image watermarking.

In classical computing, an information unit is represented by a bit (Bit), which has only two states: 0 state or 1 state. In quantum computing, the information unit is represented by a quantum bit (Qubit), which has two basic quantum states |0> and |1>, and the basic quantum state is referred to as the ground state (Basis State). A qubit can be a linear combination of two ground states, often called a superposition state (Superposition), which can be expressed as |ψ>=a|0>+b|1>. Where a and b are two complex numbers, satisfying |a| ² +|b| ² =1, so it is also called probability amplitude. The ground state |0> and |1> can be expressed as a vector:

Their dual vectors can be expressed as <0|=[1 0], <1|=[0 1].

Tensor product (tensorproduct) is a method of combining small vector spaces to form a larger vector space, using the symbol Indicates that it has the following meanings:

Suppose U is n×n and V is m×m two complex matrices

So

Suppose the set of two unitary matrices is: with There are m n×n matrices in There are n m×m matrices in . The extended tensor product is an mn×mn matrix in

when Each matrix in is the same, A ⁱ =A, at this time can be written as if at the same time Each matrix in is the same B ⁱ =B, then the expanded tensor product degenerates into an ordinary tensor product

A quantum circuit can be composed of a sequence of qubit gates. In the representation diagram of a quantum circuit, each line represents the connection of the quantum circuit, and the execution order of the quantum circuit is from left to right. Qubit gates can be conveniently represented in matrix form, and single-qubit gates can be represented by a 2×2 unitary matrix U, that is, U ⁺ U = I, where U ⁺ is the conjugate transpose matrix of U, and I is the identity matrix . Among the two-qubit gates, the most important one is the controlled NOT gate, which has two qubit input and output, which are the control qubit and the target qubit respectively. When the control bit is 1, it is represented by a black dot, and when the control bit is 0, it is represented by a white dot. The names, symbols and corresponding matrix representations of some basic qubit gates are shown in Fig. 1.

Since the existing classic D ⁽⁴⁾ wavelet transform realizes the complexity of electronic circuit design as Θ(n2 ⁿ ), it is relatively complicated and fails to meet the needs of the society. Therefore, it is necessary to design a method for realizing electronic circuit design with lower complexity.

Contents of the invention

The invention provides a method for realizing quantum circuit design by quantum D ⁽⁴⁾ wavelet transform, which solves the problem of high complexity of realizing electronic circuit design by conventional classical D ⁽⁴⁾ wavelet transform

The present invention solves the above problems through the following technical solutions:

Fully utilize the unique performance of quantum computing such as quantum parallelism and quantum superposition, use the extended tensor product, establish the iterative formula of D ⁽⁴⁾ wavelet transform, and use the quantum controlled gate to realize the extended tensor product, so as to realize the quantum D ⁽⁴⁾ Wavelet transform. Specifically, according to the principle of extended tensor product operation, two quantum D ⁽⁴⁾ wavelet transforms and two quantum D ⁽⁴⁾ wavelet inverse transforms are designed;

The realization circuit of 2 quantum D ⁽⁴⁾ wavelet transforms comprises the realization circuit of single-layer quantum D ⁽⁴⁾ wavelet transform and the realization circuit of K+1 layer quantum D ⁽⁴⁾ wavelet transform respectively;

The two realization circuits of quantum D ⁽⁴⁾ wavelet inverse transform respectively include the realization circuit of single-layer quantum D ⁽⁴⁾ wavelet inverse transformation and the realization circuit of K+1 layer quantum D ⁽⁴⁾ wavelet inverse transformation.

In the above-mentioned scheme, it is preferred that the concrete process of the realization circuit of described single-layer quantum D ⁽⁴⁾ wavelet transform is:

D ⁽⁴⁾ The wavelet kernel matrix is defined as:

in

can be expressed in tensor product as

in is the tensor product operation symbol, is the n times tensor product of I ₂ , And and are two rotation matrices,

unitary matrix The iterative formula for is as follows:

where X and _I2 are single-qubit gates in Figure 1, is the tensor product operation symbol, is the tensor product of n times of I ₂ , and the initial value of iteration is Q ₂ =X;

suppose Is a 2 ⁿ × 2 ⁿ unitary matrix, calculate the tensor product with available

From formula (3), we can get Substituting it into formula (2), thus obtaining the single-layer quantum D ⁽⁴⁾ wavelet transform:

Using the formula (5), the realization circuit of single-layer quantum D ⁽⁴⁾ wavelet transform with complexity Θ(n ² ) is designed.

In the above-mentioned scheme, it is preferred that the design process of the implementation circuit of the single-layer quantum D ⁽⁴⁾ wavelet inverse transform is:

By inverting the formula (5), the iterative formula of the D ⁽⁴⁾ wavelet kernel matrix inverse transformation and the iterative formula of the unitary matrix Q _2n can be obtained:

The initial value of iteration is (Q ₂ ) ^-1 =X;

Combined with the formula (6), the implementation circuit of the single-layer quantum D ⁽⁴⁾ wavelet inverse transform with a complexity of Θ(n ² ) is designed.

In the above-mentioned scheme, it is preferred that the realization circuit design and implementation process of the K+1 layer quantum D ⁽⁴⁾ wavelet transform is: the k+1 layer quantum D ⁽⁴⁾ wavelet transform is defined as Using the extended tensor product, we can get:

where I ₂ is the single-qubit gate in Figure 1, is the tensor product operation symbol, is the n times tensor product of I ₂ , is the k-level quantum D ⁽⁴⁾ wavelet transform, is a single-layer quantum D(4) wavelet transform, and the initial value of iteration is 1≤k≤n-2, k and n are both positive integers;

Combining formulas (5) and (7), the implementation circuit of K+1 layer quantum D ⁽⁴⁾ wavelet transform with complexity Θ(n ² ) is designed.

In the above-mentioned scheme, it is preferred that the implementation of the implementation circuit of the K+1 layer quantum D ⁽⁴⁾ wavelet inverse transform is as follows:

make is k+1 layer quantum D ⁽⁴⁾ wavelet inverse transform, inverse formula (7), can get:

where I ₂ is the single-qubit gate in Figure 1, is the tensor product operation symbol, is the n times tensor product of I ₂ , is the k-level quantum D ⁽⁴⁾ wavelet inverse transform, is the single-layer quantum D(4) wavelet inverse transform (see formula (5)), and the initial value of iteration is 1≤k≤n-2, k and n are both positive integers;

Combining formulas (6) and (8), the implementation circuit of k+1 layer quantum D ⁽⁴⁾ wavelet inverse transform with complexity Θ(n ² ) is designed.

Advantage and effect of the present invention are:

1, the present invention is compared with existing quantum D ⁽⁴⁾ wavelet transform realization technology, the present invention designs single-layer quantum D ⁽⁴⁾ wavelet transform, has used two rotation matrices to replace general unitary matrix, is to present in the method There is an innovation of quantum D ⁽⁴⁾ wavelet transform technology.

2, the present invention compares with existing quantum D ⁽⁴⁾ wavelet transform realization technology, the present invention has designed the realization circuit of multilayer quantum D ⁽⁴⁾ wavelet transform and multilayer quantum D ⁽⁴⁾ wavelet inverse transform, thereby constructs A relatively complete quantum D ⁽⁴⁾ wavelet transform system. However, the prior art only realizes the single-layer quantum D ⁽⁴⁾ wavelet transform, and the present invention is the perfection and improvement of the existing quantum D (4) wavelet transform realization technology.

3, the present invention compares with classic D ⁽⁴⁾ wavelet transform realization technology, the quantum D ⁽⁴⁾ wavelet transform that the present invention utilizes quantum circuit to realize is a kind of efficient transformation method, the quantum D ⁽⁴⁾ wavelet of the present invention's design The implementation complexity of the transform is Θ(n ² ), while the implementation complexity of the classical fast D ⁽⁴⁾ wavelet transform is Θ(2 ⁿ ).

4, the present invention has the unique properties of quantum computing such as sufficient quantum parallelism and quantum superposition, adopts the tensor product of expansion sheet, first realizes single-layer quantum D ⁽⁴⁾ wavelet transform and single-layer quantum D ⁽⁴⁾ wavelet inverse transform The iterative formula, and then establish the iterative formula of the multi-layer quantum D ⁽⁴⁾ transform and the corresponding iterative formula of the quantum D ⁽⁴⁾ wavelet inverse transform. And the quantum circuit is used to realize the quantum D ⁽⁴⁾ wavelet transform and the corresponding quantum D ⁽⁴⁾ wavelet inverse transform.

Description of drawings

Fig. 1 is the representative figure of basic quantum gate of the present invention and corresponding matrix;

Fig. 2 is the extended tensor product of the present invention And the corresponding quantum implementation circuit diagram;

Fig. 3 is the realization circuit diagram of single-layer quantum D ⁽⁴⁾ wavelet transform of the present invention;

Fig. 4 is the simplified symbol representation figure of the realization circuit of single-layer quantum D ⁽⁴⁾ wavelet transform of the present invention;

Fig. 5 is the realization circuit diagram of single-layer quantum D ⁽⁴⁾ wavelet inverse transform of the present invention;

Fig. 6 is the simplified symbol representation figure of the realization circuit of single-layer quantum D ⁽⁴⁾ wavelet inverse transform of the present invention;

Fig. 7 is the realization circuit diagram of k+1 layer quantum D ⁽⁴⁾ wavelet transform of the present invention;

Fig. 8 is the realization circuit diagram of k+1 layer quantum D ⁽⁴⁾ wavelet inverse transform of the present invention;

Fig. 9 is the realization circuit diagram of single-layer quantum D ⁽⁴⁾ wavelet transform of the present invention;

Fig. 10 is the realization circuit diagram of single-layer quantum D ⁽⁴⁾ wavelet inverse transform of the present invention;

Fig. 11 is the realization circuit diagram of two-layer quantum D ⁽⁴⁾ wavelet transform of the present invention;

Fig. 12 is the realization circuit diagram of two-layer quantum D ⁽⁴⁾ wavelet inverse transform of the present invention.

detailed description

The present invention will be further described below in conjunction with embodiment.

Example 1:

A method for realizing quantum circuit design by quantum D(4) wavelet transform, combining quantum calculation with classical D ⁽⁴⁾ wavelet transform technology to obtain quantum D ⁽⁴⁾ wavelet transform. The D ⁽⁴⁾ wavelet transform is designed according to the extended tensor product operation principle to design the realization circuit of the single-layer quantum D ⁽⁴⁾ wavelet transform.

D ⁽⁴⁾ The wavelet kernel matrix is defined as:

in

can be expressed in tensor product as

in is the tensor product operation symbol, is the n times tensor product of I ₂ , And and are two rotation matrices,

unitary matrix The iterative formula for is as follows:

where X and _I2 are single-qubit gates in Figure 1, is the tensor product operation symbol, is the tensor product of n times of I ₂ , and the initial value of iteration is Q ₂ =X;

suppose Is a 2 ⁿ × 2 ⁿ unitary matrix, calculate the tensor product with available

From formula (3), we can get Substituting it into formula (2), thus obtaining the single-layer quantum D ⁽⁴⁾ wavelet transform:

The quantum circuits corresponding to these two tensor products are shown in Figure 2. In conjunction with formula ((5), the quantum realization circuit of single-layer quantum D ⁽⁴⁾ wavelet transform is as shown in Figure 3, and its simplified symbol representation is as shown in Figure 4. Known by Figure 5, single-layer quantum D ⁽⁴⁾ wavelet transform The complexity of the quantum implementation circuit of is Θ(n ² ).

Single-layer quantum D ⁽⁴⁾ wavelet transform of the present invention's design The implementation circuit is shown in Figure 9. Substituting n=3 into formula (5), we get

The quantum circuit in Fig. 9 is obtained by realizing the formula (9).

Example 2:

A method for realizing quantum circuit design by quantum D(4) wavelet transform, combining quantum calculation with classical D ⁽⁴⁾ wavelet transform technology to obtain quantum D ⁽⁴⁾ wavelet transform. The D ⁽⁴⁾ wavelet transform is designed according to the extended tensor product operation principle to design the realization circuit of the single-layer quantum D ⁽⁴⁾ wavelet inverse transform.

By inverting the formula (5), the iterative formula of the D ⁽⁴⁾ wavelet kernel matrix inverse transformation and the iterative formula of the unitary matrix Q _2n can be obtained:

The initial value of iteration is (Q ₂ ) ^-1 =X;

Combined with formula (6), the quantum implementation circuit of the single-layer quantum D ⁽⁴⁾ wavelet inverse transform is shown in Figure 5, and its simplified symbol representation is shown in Figure 6. It is known from Fig. 5 that the complexity of the quantum realization circuit of the single-layer quantum D ⁽⁴⁾ wavelet inverse transform is Θ(n ² ).

Single-layer quantum D ⁽⁴⁾ wavelet inverse transform designed by the present invention The implementation circuit is shown in Figure 10. Substituting n=3 into formula (6), we get

The quantum circuit in Fig. 10 is obtained by realizing the formula (10).

Example 3:

A method for realizing quantum circuit design by quantum D(4) wavelet transform, combining quantum calculation with classical D ⁽⁴⁾ wavelet transform technology to obtain quantum D ⁽⁴⁾ wavelet transform. The D ⁽⁴⁾ wavelet transform is designed according to the extended tensor product operation principle to design the realization circuit of the K+1 layer quantum D ⁽⁴⁾ wavelet transform.

Define the k+1 layer quantum D ⁽⁴⁾ wavelet transform as Using the extended tensor product, we can get:

where I ₂ is the single-qubit gate in Figure 1, is the tensor product operation symbol, is the n times tensor product of I ₂ , is the k-level quantum D ⁽⁴⁾ wavelet transform, is a single-layer quantum D(4) wavelet transform, and the initial value of iteration is 1≤k≤n-2, k and n are both positive integers;

Combining formulas (5) and (7), on the basis of realizing the quantum circuit in Figure 4, the quantum realization circuit of the k+1 layer quantum D ⁽⁴⁾ wavelet transform is shown in Figure 7, and it can be seen that the complexity of the quantum realization circuit The degree is Θ(n ² ).

Two layers of quantum D ⁽⁴⁾ wavelet transform designed by the present invention The implementation circuit is shown in Figure 11. Substituting n=3 and k=1 into formula (7), we get

The quantum circuit in Fig. 11 is obtained by realizing the formula (11).

Example 4:

A method for realizing quantum circuit design by quantum D(4) wavelet transform, combining quantum calculation with classical D ⁽⁴⁾ wavelet transform technology to obtain quantum D ⁽⁴⁾ wavelet transform. Based on the extended tensor product operation principle, the D ⁽⁴⁾ wavelet transform is used to design the realization circuit of K+1 layer quantum D ⁽⁴⁾ wavelet inverse transform.

make is k+1 layer quantum D ⁽⁴⁾ wavelet inverse transform, inverse formula (7), can get:

where I ₂ is the single-qubit gate in Figure 1, is the tensor product operation symbol, is the n times tensor product of I ₂ , is the k-level quantum D ⁽⁴⁾ wavelet inverse transform, is the single-layer quantum D(4) wavelet inverse transform (see formula (5)), and the initial value of iteration is 1≤k≤n-2, k and n are both positive integers;

Combining formulas (6) and (8), on the basis of realizing the quantum circuit in Figure 6, the quantum realization circuit of k+1 layer quantum D ⁽⁴⁾ wavelet inverse transform is shown in Figure 8, and it can be known that the quantum realization circuit The complexity is Θ(n ² ).

Two-layer quantum D ⁽⁴⁾ wavelet inverse transform designed by the present invention The implementation circuit is shown in Figure 12. Substituting n=3 and k=1 into formula (8), we get

The quantum circuit in Fig. 12 is obtained by realizing the formula (12).

The present invention fully possesses the unique properties of quantum computing such as quantum parallelism and quantum superposition, and adopts the tensor product of expansion tensor to first realize the iterative formula of single-layer quantum D ⁽⁴⁾ wavelet transform and single-layer quantum D ⁽⁴⁾ wavelet inverse transform , and then establish the iterative formula of the multi-layer quantum D ⁽⁴⁾ transform and the corresponding iterative formula of the quantum D ⁽⁴⁾ wavelet inverse transform. And the quantum circuit is used to realize the quantum D ⁽⁴⁾ wavelet transform and the corresponding quantum D ⁽⁴⁾ wavelet inverse transform.

The preferred embodiments of the present invention have been described in detail above, but the present invention is not limited to the embodiments. Those skilled in the art can also make various equivalent modifications or replacements without violating the spirit of the present invention. , these equivalent modifications or replacements are included within the 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).