CN103971328A

CN103971328A - Storage, design and implementation method for multi-dimensional quantum gray image and multi-dimensional quantum colorful image

Info

Publication number: CN103971328A
Application number: CN201410186003.9A
Authority: CN
Inventors: 黎海生; 周日贵; 刘志强; 喻友文; 周佳丽
Original assignee: East China Jiaotong University
Current assignee: East China Jiaotong University
Priority date: 2014-05-05
Filing date: 2014-05-05
Publication date: 2014-08-06

Abstract

The storage design and implementation method of multi-dimensional quantum grayscale and color images discloses a new storage method for multidimensional quantum grayscale images based on gauge arbitrary superposition states and a storage method for multidimensional quantum color images based on three-component gauge arbitrary superposition states Storage methods, and blueprints for the quantum circuits that implement them. These design diagrams use basic qubit gates (including qubit controlled gates and single qubit gates) to construct storage implementation circuits for NASS-based multi-dimensional quantum grayscale images and NASSTC-based multi-dimensional quantum color images. On the one hand, the invention solves the problem of quantum image "how to store the image in the quantum system", on the other hand, it is also an efficient image compression method, which can be used in many practical image processing application fields that require multi-dimensional image compression. The attached picture of the abstract shows the storage implementation circuit based on NASS multi-dimensional quantum grayscale image.

Description

Storage design and implementation method of multi-dimensional quantum grayscale and color images

技术领域technical field

本发明涉及多维量子灰度和彩色图像的存储设计方法与实现线路，属于量子图像处理技术领域。The invention relates to a storage design method and realization circuit of multi-dimensional quantum grayscale and color images, and belongs to the technical field of quantum image processing.

背景技术Background technique

量子计算机有不同的结构模型，例如量子图灵机模型，量子线路模型，细胞自动机模型等。量子线路模型比其它的几种模型更容理解，但功能是等价的，因此采用量子线路模型来定义量子计算机：是由包含连线和基本量子门排列起来、形成的处理量子信息的量子线路建造的。量子计算机具有独特的处理数据能力，可解决现有经典计算机难以解决的数学问题，例如大数的质因子分解和离散对数求解，因此，它成为世界各国战略竞争焦点，比如，美国仿照当年成功制造原子弹的曼哈顿计划(Manhattan project)，在2009年启动了微型曼哈顿计划(Mini-Manhattan project)，投巨资去研发量子芯片。Quantum computers have different structural models, such as quantum Turing machine model, quantum circuit model, cellular automata model, etc. The quantum circuit model is easier to understand than other models, but the functions are equivalent. Therefore, the quantum circuit model is used to define the quantum computer: it is a quantum circuit composed of wiring and basic quantum gates arranged to process quantum information. built. Quantum computers have a unique ability to process data and can solve mathematical problems that are difficult for existing classical computers, such as the prime factorization of large numbers and the solution of discrete logarithms. Therefore, it has become the focus of strategic competition around the world. The Manhattan project, which built the atomic bomb, launched the Mini-Manhattan project in 2009, investing huge sums of money to develop quantum chips.

将量子计算和图像处理处理技术向结合，这种新的不同学科的交叉技术定义为量子图像处理。Combining quantum computing and image processing technology, this new interdisciplinary technology is defined as quantum image processing.

在经典计算中，信息单元用比特(Bit)表示，它只有两个状态：0态或1态。在量子计算中，信息单元用量子比特(Qubit)表示，它有两个基本量子态|0>和|1>，基本量子态简称为基态(Basis State)。任何双能级的量子系统都可用来实现量子比特，例如氢原子中的电子的基态和激发态、质子自旋在任意方向的+1/2分量和-1/2、圆偏振光的左旋和右旋等都可以分别用|0>和|1>表示。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). Any two-level quantum system can be used to realize qubits, such as the ground state and excited state of electrons in hydrogen atoms, the +1/2 component and -1/2 of proton spin in any direction, the left-handed sum of circularly polarized light Right-handed and so on can be represented by |0> and |1> respectively.

一个量子比特可以是两个基态的线性组合，常被称为叠加态(Superposition)，可表示为|ψ>＝a|0>+b|1>。其中a和b是两个复数，满足|a|²+|b|²＝1，因此也被称为概率幅。在测量量子比特时，量子态|ψ>以|a|²的概率坍缩(Collapsing)成|0>，以|b|²的概率坍缩成|1>。所以一个量子比特可以同时包含|0>和|1>的信息，这与经典计算中的比特截然不同。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. When measuring a qubit, the quantum state |ψ> collapses into |0> with the probability of |a| ² and collapses into |1> with the probability of |b| ² . So a qubit can contain both |0> and |1> information, which is very different from a bit in classical computing.

张量积(Tensor Product)是将小的向量空间合在一起，构成更大向量空间的一种方法，用符号表示，它有如下的含义：Tensor Product (Tensor Product) is a method of combining small vector spaces to form a larger vector space, using the symbol Indicates that it has the following meanings:

假设U和V是两个复矩阵Suppose U and V are two complex matrices

$U u = = [\begin{matrix} {u u}_{0000} & {u u}_{0101} \\ {u u}_{1010} & {u u}_{1111} \end{matrix}],,$ $V V = = [\begin{matrix} {v v}_{0000} & {v v}_{0101} \\ {v v}_{1010} & {v v}_{1111} \end{matrix}]$

那么So

$U u &CircleTimes; &CircleTimes; V V = = [\begin{matrix} {u u}_{0000} {v v}_{0000} & {u u}_{0000} {v v}_{0101} & {u u}_{0101} {v v}_{0000} & {u u}_{0101} {v v}_{0101} \\ {u u}_{0000} {v v}_{1010} & {u u}_{0000} {v v}_{1111} & {u u}_{0101} {v v}_{1010} & {u u}_{0101} {v v}_{1111} \\ {u u}_{1010} {v v}_{0000} & {u u}_{1010} {v v}_{0101} & {u u}_{1111} {v v}_{0000} & {u u}_{1111} {v v}_{0101} \\ {u u}_{1010} {v v}_{1010} & {u u}_{1010} {v v}_{1111} & {u u}_{1111} {v v}_{1010} & {u u}_{1111} {v v}_{1111} \end{matrix}]$

对于两个基态|u>和|v>，它们的张量积常用缩写符号|uv>，|u>|v>或|u,v>表示，例如对于基态|0>和|1>，它们的张量积可表示为For two ground states |u> and |v>, their tensor product Commonly used abbreviations |uv>, |u>|v> or |u, v>, for example, for the ground state |0> and |1>, their tensor product can be expressed as

$| | 00 > > &CircleTimes; &CircleTimes; | | 11 > > = = | | 00 > > | | 11 > > = = | | 0101 > > = = \begin{matrix} \end{matrix} [\begin{matrix} 11 \\ 00 \end{matrix}] &CircleTimes; &CircleTimes; [\begin{matrix} 00 \\ 11 \end{matrix}] = = [\begin{matrix} 00 \\ 11 \\ 00 \\ 00 \end{matrix}]$

对于矩阵U的n次张量积可简写成对于量子态|u>的n次张量积也可简写成 For n tensor products of matrix U Can be abbreviated as For the n times tensor product of quantum state |u> can also be abbreviated as

一个双量子比特可由两个单量子比特张量运算合成，它有四个基态|00>、|01>、|10>和|11>。因此，一个双量子比特的状态可描述为A two-qubit can be synthesized by two single-qubit tensor operations, and it has four ground states |00>, |01>, |10> and |11>. Therefore, the state of a two-qubit can be described as

|ψ>＝a₀₀|00>+a₀₁|01>+a₁₀|10>+a₁₁|11>|ψ>＝a ₀₀ |00>+a ₀₁ |01>+a ₁₀ |10>+a ₁₁ |11>

其中测量结果|00>、|01>、|10>和|11>出现的概率分别是|a₀₀|²、|a₀₁|²、|a₁₀|²和|a₁₁|²，并且满足归一化条件|a₀₀|²+|a₀₁|²+|a₁₀|²+|a₁₁|²＝1。The probabilities of the measurement results |00>, |01>, |10> and |11> are |a ₀₀ | ² , |a ₀₁ | ² , |a ₁₀ | ² and |a ₁₁ ^| The normalization condition |a ₀₀ | ² +|a ₀₁ | ² +|a ₁₀ | ² +|a ₁₁ | ² =1.

若一个量子系统由n量子比特构成，这个量子系统有2ⁿ个相互正交的基态|i₁i₂…i_n>,i₁,i₂,…,i_n∈{0,1}，这2ⁿ个基态张成一个2ⁿ维Hilbert空间，则该量子系统的状态可表示为If a quantum system consists of n qubits, and this quantum system has 2 ⁿ mutually orthogonal ground states |i ₁ i ₂ …i _n >,i ₁ ,i ₂ ,…,i _n ∈{0,1}, this 2 ⁿ ground states form a 2 ^n- dimensional Hilbert space, then the state of the quantum system can be expressed as

$| | ψ ψ > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {a a}_{i i} | | i i > >$

其中i＝i₁i₂…i_n是整数i的二进制展开，并且满足归一化条件 Where i=i ₁ i ₂ ...i _n is the binary expansion of the integer i, and satisfies the normalization condition

如果用一组量子逻辑门组成的量子线路可以以任意精度逼近任意的酉运算，那么这组量子门就是通用的。重要的一类通用门是单量子比特门和受控非门，即一般的量子逻辑门可以由单量子比特门和受控非门构成，一个具体的例子是Hadamard门、相位门、π/8门和受控非门是通用的。If a quantum circuit composed of a group of quantum logic gates can approximate any unitary operation with arbitrary precision, then this group of quantum gates is universal. An important class of general-purpose gates is a single-qubit gate and a controlled NOT gate, that is, a general quantum logic gate can be composed of a single-qubit gate and a controlled NOT gate. A specific example is Hadamard gate, phase gate, π/8 Gates and controlled NOT gates are generic.

量子比特门可以方便的用矩阵形式表示，单量子比特门可以用一个2×2的酉矩阵U表示，即U+U＝I，其中U⁺是U的共轭转置矩阵，I是单位阵。单量子比特门表示见图1，其中U是一个2×2的酉矩阵。将一个具体矩阵代替图1中的U矩阵，就可以得到一个具体的单量子比特门的符号表示，常用的单量子比特门的名称、符号及相应的矩阵表示见图2。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 . The single-qubit gate representation is shown in Figure 1, where U is a 2×2 unitary matrix. By replacing the U matrix in Figure 1 with a specific matrix, a specific symbolic representation of a single-qubit gate can be obtained. The names, symbols and corresponding matrix representations of commonly used single-qubit gates are shown in Figure 2.

在双量子比特门中，最重要是受控U门，U是一个任意2×2的酉矩阵，它有两个量子的比特输入和输出，分别是控制量子比特和目标量子比特。当控制位为1时，将这个受控U门命名为U_C1，当控制位为0时，将这个受控U门命名为U_C0，这两个受控U门的名称、符号及相应的矩阵表示见图3。Among the two-qubit gates, the most important one is the controlled U gate. U is an arbitrary 2×2 unitary matrix, which has two quantum bit inputs and outputs, which are the control qubit and the target qubit, respectively. When the control bit is 1, the controlled U gate is named U _C1 , when the control bit is 0, the controlled U gate is named U _C0 , the names, symbols and corresponding See Figure 3 for matrix representation.

将U_C1和U_C0分别作用在量子态|φ>＝c|0>+d|1>和|ψ>＝α|0>+β|1>上，结果是Applying U _C1 and U _C0 to the quantum states |φ>=c|0>+d|1> and |ψ>=α|0>+β|1> respectively, the result is

U_C1(|φ>|ψ>)＝c|0>|ψ>+d|1>(U|ψ>)U _C1 (|φ>|ψ>)＝c|0>|ψ>+d|1>(U|ψ>)

和and

U_C0(|φ>|ψ>)＝c|0>(U|ψ>)+d|1>|ψ>U _C0 (|φ>|ψ>)＝c|0>(U|ψ>)+d|1>|ψ>

其中U|ψ>的含义为where U|ψ> means

U|ψ>＝(αu₀₀+βu₀₁)|0+(αu₁₀+βu₁₁)|1>U|ψ>＝(αu ₀₀ +βu ₀₁ )|0+(αu ₁₀ +βu ₁₁ )|1>

如果U_C1和U_C0的U矩阵是图2中的X矩阵，那么这两个特殊的受控门被称为受控非门，简记为N_C1和N_C0，它们的名称、符号及相应的矩阵表示见图3。If the U matrix of U _C1 and U _C0 is the X matrix in Figure 2, then these two special controlled gates are called controlled NOT gates, abbreviated as N _C1 and N _C0 , their names, symbols and corresponding The matrix representation of is shown in Figure 3.

将N_C1和N_C0分别作用在量子态|φ>＝c|0>+d|1>和|ψ>＝α|0>+β|1>上，得到Applying N _C1 and N _C0 to the quantum states |φ>=c|0>+d|1> and |ψ>=α|0>+β|1> respectively, we get

N_C1(|φ>|ψ>)＝c|0>|ψ>+d|1>(X|ψ>)N _C1 (|φ>|ψ>)＝c|0>|ψ>+d|1>(X|ψ>)

和and

N_C0(|φ>|ψ>)＝c|0>(X|ψ>)+d|1>|ψ>N _C0 (|φ>|ψ>)＝c|0>(X|ψ>)+d|1>|ψ>

其中X|ψ>的含义为where X|ψ> means

X|ψ>＝X(α|0>+β|1>)＝β|0>+α|1>X|ψ>=X(α|0>+β|1>)=β|0>+α|1>

设U是一个任意2×2的酉矩阵，将n(n≥2)量子比特受控门命名为Cⁿ(U)，分别有(n-1)个控制量子比特，1个目标量子比特，并假定二进制数i₁,i₂,…,i_n-1分别是(n-1)个控制位上的数字，则Cⁿ(U)的符号表示如图5所示。Let U be an arbitrary 2×2 unitary matrix, and name the controlled gate of n(n≥2) qubits as C ⁿ (U), there are respectively (n-1) control qubits and 1 target qubit, And assuming that the binary numbers i ₁ , i ₂ ,...,i _n-1 are numbers on (n-1) control bits respectively, the symbol representation of C ⁿ (U) is shown in FIG. 5 .

将图5中的Cⁿ(U)门作用到n个单量子比特的量子态|x₁x₂…x_n-1>|ψ>上，可得到Applying the C ⁿ (U) gate in Figure 5 to the quantum state |x ₁ x ₂ …x _n-1 >|ψ> of n single qubits, we can get

${C C}^{n no} ((U u)) ((| | {x x}_{11} {x x}_{22} . . . . . . {x x}_{n no - - 11} > > | | ψ ψ > >)) = = | | {x x}_{11} . . . . . . {x x}_{n no - - 11} > > {U u}^{r r (({x x}_{11} . . . . . . {x x}_{n no - - 11},, {i i}_{11} . . . . . . {i i}_{n no - - 11}))} | | ψ ψ > >$

其中如果x₁…x_n-1＝i₁…i_n-1，则函数f(x₁…x_n-1,i₁…i_n-1)为1，否则f(x₁…x_n-1,i₁…i_n-1)为0，并令U⁰＝I,U¹＝U。Wherein if x ₁ …x _n-1 ＝i ₁ …i _n-1 , then the function f(x ₁ …x _n-1 ,i ₁ …i _n-1 ) is 1, otherwise f(x ₁ …x _{n- 1} , i ₁ . . . i _n-1 ) is 0, and U ⁰ =I, U ¹ =U.

需要说明的是，在量子线路的表示图中，每条线都表示量子线路的连线，量子线路的执行顺序是从左到右。It should be noted that, in the representation diagram of the quantum circuit, each line represents the connection of the quantum circuit, and the execution order of the quantum circuit is from left to right.

发明内容Contents of the invention

本发明的目的是，设计一种新型的多维量子灰度和彩色图像的存储方法，并实现它们的量子线路设计图。这些设计图用基本的量子比特门(包括量子比特受控门和单量子比特门)，分别构建了基于规范任意的叠加态(Normal ArbitrarySuperposition State,NASS)的多维量子灰度图像的存储的实现线路和基于三分量的规范任意的叠加态(Normal Arbitrary Superposition State with Three Components,NASSTC)多维量子彩色图像的存储的实现线路。The purpose of the present invention is to design a novel storage method for multi-dimensional quantum grayscale and color images, and realize their quantum circuit design diagrams. These design diagrams use basic qubit gates (including qubit-controlled gates and single-qubit gates) to build implementation circuits for storing multi-dimensional quantum grayscale images based on Normal Arbitrary Superposition State (NASS). And the implementation circuit of the storage of the multi-dimensional quantum color image based on the three-component standard arbitrary superposition state (Normal Arbitrary Superposition State with Three Components, NASSTC).

实现本发明目的的指导思想是，本发明充分发挥量子并行性和量子叠加性等量子计算的独特性能，利用量子线路来实现基于NASS的多维量子灰度图像和基于NASSTC的多维量子彩色图像的存储。The guiding ideology for realizing the purpose of the present invention is that the present invention fully exploits the unique properties of quantum computing such as quantum parallelism and quantum superposition, and uses quantum circuits to realize the storage of multi-dimensional quantum grayscale images based on NASS and multi-dimensional quantum color images based on NASSTC .

本发明的技术方案是，本发明方法采用n量子比特受控门Cⁿ(U)实现规范任意的叠加态(NASS)量子态，从而实现多维量子灰度图像的存储；所述方法在实现NASS量子态的基础上，利用1个RGB彩色可以用三个灰度值表示的特性，综合使用三个NASS量子态的实现线路来实现三分量的规范任意的叠加态(NASSTC)量子态，从而实现多维量子彩色图像的存储。The technical scheme of the present invention is, the inventive method adopts n qubit controlled gate C ⁿ (U) to realize canonical arbitrary superposition state (NASS) quantum state, thereby realizes the storage of multi-dimensional quantum gray scale image; Described method realizes NASS On the basis of the quantum state, using the characteristic that one RGB color can be represented by three gray values, the implementation circuit of three NASS quantum states is used comprehensively to realize the three-component canonical arbitrary superposition state (NASSTC) quantum state, thereby realizing Storage of multidimensional quantum color images.

所述多维量子灰度图像的存储实现方法以量子线路模型为基础，设计基于NASS的多维量子灰度图像存储实现方法。The storage realization method of the multidimensional quantum grayscale image is based on the quantum circuit model, and a NASS-based storage realization method of the multidimensional quantum grayscale image is designed.

所述多维量子彩色图像的存储实现方法以量子线路模型为基础，设计基于NASSTC的多维量子彩色图像的存储实现方法。The storage implementation method of the multi-dimensional quantum color image is based on the quantum circuit model, and the storage implementation method of the multi-dimensional quantum color image based on NASSTC is designed.

本发明方法的具体步骤为：The concrete steps of the inventive method are:

1、基于NASS的多维量子灰度图像的存储的设计与实现方法1. The design and implementation method of NASS-based multi-dimensional quantum grayscale image storage

(1)灰度的表示(1) Representation of grayscale

步骤1：对256个颜色排序，生成一个有序颜色集合Color＝{color₁,color₂,…,color₂₅₆}，颜色按灰度值升序排列，color_i是灰度值为i-1的颜色。Step 1: Sort the 256 colors to generate an ordered color set Color={color ₁ ,color ₂ ,...,color ₂₅₆ }, the colors are arranged in ascending order of grayscale value, color _i is the color with grayscale value i-1 .

步骤2：本发明定义一个有序角度集φ＝{φ₁,φ₂,…,φ₂₅₆}Step 2: The present invention defines an ordered angle set φ={φ ₁ ,φ ₂ ,…,φ ₂₅₆ }

${φ φ}_{i i} = = \frac{π π ((i i - - 11))}{22 ((256256 - - 11))}$

其中i∈{1,2,…,256}，并创建颜色与角度一一对应的双射函数F₁ where i∈{1,2,…,256}, and create a bijective function F ₁ with one-to-one correspondence between colors and angles

F₁:Color□φ (1)F ₁ : Color φ (1)

其中Color是步骤1中已经排好序的颜色集，F₁(color_i)＝φ_i，F₁ ^-1(φ_i)＝color_i。Where Color is the sorted color set in step 1, F ₁ (color _i )=φ _i , F ₁ ^-1 (φ _i )=color _i .

例如，φ₁和φ₂₅₆分别对应灰度值为0和255的颜色。本发明定义灰度色彩的距离为：For example, φ ₁ and φ ₂₅₆ correspond to colors with grayscale values of 0 and 255, respectively. The present invention defines the distance of the gray scale color as:

D_G(color_i,color_j)＝|i-j|D _G (color _i ,color _j )＝|ij|

因此，可推导出Therefore, it can be deduced

$| | {φ φ}_{i i} - - {φ φ}_{j j} | | = = \frac{π π}{22 ((256256 - - 11))} {D D.}_{G G} ((colo colo {r r}_{i i} - - {color color}_{j j}))$

即，相邻的两个角度φ_i和φ_j对应的颜色color_i和color_j是相近的，反之，相近的两种颜色color_i和color_j对应的角度φ_i和φ_j是相邻的。That is, the colors color _i and color _j corresponding to two adjacent angles φ _i and φ _j are similar, and conversely, the angles φ _i and φ _j corresponding to two similar colors color _i and color _j are adjacent.

(2)基于NASS的多维量子灰度图像的存储表示(2) NASS-based storage representation of multi-dimensional quantum grayscale images

假设V是由正交基向量b₁,b₂,…b_k张成的k维欧式的向量空间，那么k维数字灰度图像可以用函数f:V→C表示，其中V可表示图像的位置信息，C是颜色集，f(V)是图像的位置对应的像素的颜色集。在本发明中令通过有序颜色集Color与有序角度集φ之间的双射函数F₁:Color□φ，见公式(1)，就可以用φ表示k维数字图像的颜色集。Assuming that V is a k-dimensional Euclidean vector space spanned by orthogonal basis vectors b ₁ , b ₂ ,…b _k , then a k-dimensional digital grayscale image can be expressed by the function f:V→C, where V can represent the Position information, C is the color set, f(V) is the color set of the pixel corresponding to the position of the image. In the present invention order Through the bijective function F ₁ :Color□φ between the ordered color set Color and the ordered angle set φ, see formula (1), φ can be used to represent the color set of a k-dimensional digital image.

2ⁿ维Hilbert空间的一个任意的量子叠加态可表示如下An arbitrary quantum superposition state of 2 ^n- dimensional Hilbert space can be expressed as follows

$| | {ψ ψ}_{a a} > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {a a}_{i i} | | i i > > - - - - - - ((22))$

其中{|1>,…,|2ⁿ-1>}是2ⁿ维Hilbert空间的一组正交基，a_i是一个任意实数。Where {|1>,...,|2 ⁿ -1>} is a set of orthogonal basis of 2 ⁿ -dimensional Hilbert space, and a _i is an arbitrary real number.

为了表示一幅k维数字图像，(2)被改写为To represent a k-dimensional digital image, (2) is rewritten as

$| | {ψ ψ}_{φk φk} > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {a a}_{i i} | | i i > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {a a}_{i i} | | {v v}_{11} > > | | {v v}_{22} > > . . . . . . | | {v v}_{k k} > > . . . . . . ((33))$

其中i＝i₁…i_ji_j+1…i_l…i_m…i_n、v₁＝i₁…i_j、v₂＝i_j+1…i_l和v_k＝i_m…i_n分别整数i、v₁、v₂和v_k的二进制展开；|i>＝|v₁>|v₂>…|v_k>是n量子比特张量积|i₁…i_ji_j+1…i_l…i_m…i_n>的简化表示，可以表示k维空间的坐标(v₁,v₂,…,v_k)；a_i∈φ，φ是有序角度集，见公式(1)，这样a_i表示坐标|i>对应像素的颜色。Where i = i ₁ ...i _j i _j+1 ...i _l ...i _m ...i _n , v ₁ =i ₁ ...i _j , v ₂ =i _j+1 ...i _l and v _k = _im ...i _n The binary expansion of the integers i, v ₁ , v ₂ and v _k respectively; |i>=|v ₁ >|v ₂ >…|v _k > is the tensor product of n qubits |i ₁ …i _j i _j+1 The simplified representation of …i _l …i _m …i _n > can represent the coordinates (v ₁ ,v ₂ ,…,v _k ) of the k-dimensional space; a _i ∈ φ, φ is an ordered angle set, see the formula (1 ), so that a _i represents the color of the corresponding pixel at coordinates |i>.

为了归一化公式(3)中的量子态|ψ_φk>，本发明令In order to normalize the quantum state |ψ _φk > in formula (3), the present invention makes

$\{\begin{matrix} {G G}_{φ φ} = = \sqrt{{Σ Σ}_{y the y = = 00}^{22^{n no} - - 11} {a a}_{y the y}^{22}} \\ {θ θ}_{i i} = = \frac{{a a}_{i i}}{{G G}_{φ φ}} \end{matrix} - - - - - - ((44))$

将公式(4)中的θ_i代替公式(3)中的a_i，本发明得到一个NASS态|ψ_k>Replacing θ _i in formula (4) with a _i in formula (3), the present invention obtains a NASS state |ψ _k >

$| | {ψ ψ}_{k k} > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {θ θ}_{i i} | | i i > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {θ θ}_{i i} | | {v v}_{11} > > | | {v v}_{22} > > . . . . . . | | {v v}_{k k} > > - - - - - - ((55))$

其中 $(Σ_{i = 0}^{2^{n} - 1} θ_{i}^{2}) = 1 .$ in $(Σ_{i = 0}^{2^{no} - 1} θ_{i}^{2}) = 1 .$

对于一幅具体的图像，公式(4)中的G_φ是一个常量，从公式(3)知，a_i表示坐标|i>对应像素的颜色，结合公式(4)，可以通过创建一个双射函数F₃:θ_i□a_i，使θ_i和a_i表示相同的颜色。因此，公式(5)中的NASS态|ψ_k>能表示一幅k维图像。For a specific image, G _φ in formula (4) is a constant. From formula (3), a _i represents the color of the corresponding pixel with coordinate |i>, combined with formula (4), you can create a bijection Function F ₃ : θ _i □ a _i , so that θ _i and a _i represent the same color. Therefore, the NASS state |ψ _k > in formula (5) can represent a k-dimensional image.

为了使公式(5)更明确，考虑dim(|v_i>)＝m_i，i＝1,2,…,k，作为一个例子。这样，一个表示2ⁿ个像素的k维灰度图像的NASS态可描述如下To make formula (5) more explicit, consider dim(|v _i >)=m _i , i=1, 2, . . . , k, as an example. Thus, a NASS state representing a k-dimensional grayscale image of 2 ⁿ pixels can be described as follows

其中1≤j≤k，整数i的二进制展开为Where 1≤j≤k, the binary expansion of the integer i is

$i i = = {i i}_{11} . . . . . . {i i}_{{m m}_{11}} . . . . . . {i i}_{(({Σ Σ}_{h h = = 11}^{j j - - 11} {m m}_{h h})) + + 11} . . . . . . {i i}_{(({Σ Σ}_{h h = = 11}^{j j - - 11} {m m}_{h h})) + + {m m}_{j j}} . . . . . . {i i}_{(({Σ Σ}_{h h = = 11}^{k k - - 11} {m m}_{h h})) + + 11} . . . . . . {i i}_{(({Σ Σ}_{h h = = 11}^{k k - - 11} {m m}_{h h})) + + {m m}_{k k}}$

将n＝5、k＝3、m₁＝2、m₂＝2和m₃＝1代入公式(6)，得到Substituting n=5, k=3, m ₁ =2, m ₂ =2 and m ₃ =1 into formula (6), we get

$\begin{matrix} | | {ψ ψ}_{33} > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {θ θ}_{i i} | | {i i}_{11} {i i}_{22} > > | | {i i}_{33} {i i}_{44} > > | | {i i}_{55} > > \\ = = {θ θ}_{00} | | 0000 > > | | 0000 > > | | 00 > > + + {θ θ}_{11} | | 0000 > > | | 0000 > > | | 11 > > + + . . . . . . + + {θ θ}_{3030} | | 1111 > > | | 1111 > > | | 00 > > + + {θ θ}_{3131} | | 1111 > > | | 1111 > > | | 11 > > \end{matrix}$

因此|ψ₃>能表示一幅4×4×2的3维图像，如图5所示。Therefore |ψ ₃ > can represent a 4×4×2 3D image, as shown in Figure 5.

(3)基于NASS的多维量子灰度图像的存储实现方法(3) Storage implementation method of multi-dimensional quantum grayscale image based on NASS

一个酉算子被本发明定义为A unitary operator is defined by the present invention as

${R R}_{x x} ((θ θ)) = = [\begin{matrix} cos cos θ θ & sin sin θ θ \\ sin sin θ θ & - - cos cos θ θ \end{matrix}] - - - - - - ((77))$

其中θ∈[0，2π]，并且酉算子R_x(θ)有如下性质：where θ∈[0, 2π], and the unitary operator R _x (θ) has the following properties:

$\{\begin{matrix} {R R}_{x x} ((θ θ)) \cdot \cdot {R R}_{x x} ((θ θ)) = = I I \\ {R R}_{x x} ((00)) = = Z Z \\ {R R}_{x x} ((\frac{π π}{44})) = = H h \\ {R R}_{x x} ((\frac{π π}{22})) = = X x \end{matrix}$

其中I、Z、H和X分别是图2中的单位算子I、Pauli-Z算子、Hadmard算子和Pauli-X算子。Among them, I, Z, H and X are the unit operator I, Pauli-Z operator, Hadmard operator and Pauli-X operator in Fig. 2 respectively.

一个角度序列被给出：A sequence of angles is given:

$\{\begin{matrix} {α α}_{j j} = = arctan arctan \sqrt{\frac{{Σ Σ}_{{i i}_{22} . . . . . . {i i}_{n no}} {| | {θ θ}_{11 {i i}_{22} . . . . . . {i i}_{n no}} | |}^{22}}{{Σ Σ}_{{i i}_{22} . . . . . . {i i}_{n no}} {| | {θ θ}_{00 {i i}_{22} . . . . . . {i i}_{n no}} | |}^{22}}} & j j = = 11 \\ {α α}_{j j,, {i i}_{11} . . . . . . {i i}_{j j - - 11}} = = arctan arctan \sqrt{\frac{{Σ Σ}_{{i i}_{j j + + 11} . . . . . . {i i}_{n no}} {| | {θ θ}_{{i i}_{11} . . . . . . {i i}_{j j - - 11} 11 {i i}_{j j + + 11} . . . . . . {i i}_{n no}} | |}^{22}}{{Σ Σ}_{{i i}_{j j + + 11} . . . . . . {i i}_{n no}} {| | {θ θ}_{{i i}_{11} . . . . . . {i i}_{j j - - 11} 00 {i i}_{j j + + 11} . . . . . . {i i}_{n no}} | |}^{22}}} & 22 \leq \leq j j \leq \leq n no \end{matrix} - - - - - - ((88))$

其中，假设0i₂…i_n、1i₂…i_n、i₁…i_j-10i_j+1…i_n和i₁…i_j-11i_j+1…i_n分别是整数g、h、x和y的二进制展开，那么 $θ_{{0 i}_{2} . . . i_{n}} = θ_{g},$ $θ_{{1 i}_{2} . . . i_{n}} = θ_{h},$ $θ_{i_{1} . . . i_{j - 1} 0 i_{j + 1} . . . i_{n}} = θ_{x}$ 和 $θ_{i_{1} . . . i_{j - 1} 1 i_{j + 1} . . . i_{n}} = θ_{y}$ 分别是公式(5)中量子叠加态|ψ_k>中的项|g>、|h>、|x>和|y>的系数。i₁…i_j-1是整数z的二进制展开，z＝0,1,…,(2^j-1-1)。Among them, it is assumed that 0i ₂ ...i _n , 1i ₂ ...i _n , i ₁ ...i _j-1 0i _j+1 ...i _n and i ₁ ...i _j-1 1i _j+1 ...i _n are integers g and h respectively , the binary expansion of x and y, then $θ_{{0 i}_{2} . . . i_{no}} = θ_{g},$ $θ_{{1 i}_{2} . . . i_{no}} = θ_{h},$ $θ_{i_{1} . . . i_{j - 1} 0 i_{j + 1} . . . i_{no}} = θ_{x}$ and $θ_{i_{1} . . . i_{j - 1} 1 i_{j + 1} . . . i_{no}} = θ_{the y}$ are the coefficients of the terms |g>, |h>, |x> and |y> in the quantum superposition state |ψ _k > in formula (5), respectively. i ₁ ...i _j-1 is the binary expansion of the integer z, z=0,1,...,(2 ^j-1 -1).

当j≥2时，本发明定义受控R_xji算子为When j≥2, the present invention defines the controlled R _xji operator as

${R R}_{xji xji} = = (({Σ Σ}_{h h = = 00,, h h &NotEqual; &NotEqual; i i}^{22^{j j - - 11} - - 11} | | h h > > < < h h | |)) &CircleTimes; &CircleTimes; I I + + | | i i > > < < i i | | &CircleTimes; &CircleTimes; {R R}_{x x} (({α α}_{j j,, i i})) - - - - - - ((99))$

其中0≤i≤2^j-1、i＝i₁…i_j-1是整数i的二进制展开，角度见公式(8)。由于R_x(α_j,i)是酉算子，将R_x(α_j,i)算子代替图4中的U算子，得到受控R_xji算子线路实现，如图6所示。Where 0≤i≤2 ^j -1, i=i ₁ ...i _j-1 is the binary expansion of the integer i, the angle See formula (8). Since R _x (α _j,i ) is a unitary operator, replace the U operator in Figure 4 with the R _x (α _j,i ) operator, and obtain the circuit realization of the controlled R _xji operator, as shown in Figure 6.

本发明定义酉算子R_xj为The present invention defines unitary operator R _xj as

$\{\begin{matrix} {R R}_{xj xj} = = {R R}_{x x} (({α α}_{11})) &CircleTimes; &CircleTimes; {I I}^{&CircleTimes; &CircleTimes; ((n no - - 11))} & j j = = 11 \\ {R R}_{xj xj} = = {Π Π}_{i i = = 00}^{22^{j j - - 11} - - 11} (({R R}_{xji xji} &CircleTimes; &CircleTimes; {I I}^{&CircleTimes; &CircleTimes; ((n no - - j j))})) & 22 \leq \leq j j \leq \leq n no \end{matrix} - - - - - - ((1010))$

依次将酉算子R_xj作用在初态上，本发明就可以得到一个NASS态In turn, the unitary operator R _xj acts on the initial state On, the present invention just can obtain a NASS state

$| | {ψ ψ}_{k k} > > = = (({Π Π}_{j j = = 11}^{n no} {R R}_{xj xj})) {| | 00 > >}^{&CircleTimes; &CircleTimes; n no} = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {θ θ}_{i i} | | i i > > = = {Σ Σ}_{i i = = 00}^{22^{n no} - - 11} {θ θ}_{i i} | | {v v}_{11} > > | | {v v}_{22} > > . . . . . . | | {v v}_{k k} > >$

其实现线路如图7所示，从而实现了基于NASS的多维量子灰度图像的存储。Its implementation circuit is shown in Figure 7, thereby realizing the storage of multi-dimensional quantum grayscale images based on NASS.

2、基于NASSTC的多维量子彩色图像的存储的设计与实现方法2. Design and implementation method of multi-dimensional quantum color image storage based on NASSTC

(1)基于NASSTC的多维量子彩色图像的存储表示(1) Storage representation of multi-dimensional quantum color images based on NASSTC

假设向量(y₁,y₂,y₃)表示坐标|j>对应的像素的RGB颜色，其中y₁、y₂和y₃是RGB颜色对应的三个分量的灰度值。本发明能用公式(1)计算三个角度r_j、g_j和b_j，即，Assume that the vector (y ₁ , y ₂ , y ₃ ) represents the RGB color of the pixel corresponding to the coordinate |j>, where y ₁ , y ₂ and y ₃ are the grayscale values of the three components corresponding to the RGB color. The present invention can use formula (1) to calculate three angles r _j , g _j and b _j , namely,

$\{\begin{matrix} {r r}_{j j} = = \frac{{y the y}_{11} π π}{22 ((256256 - - 11))} \\ {g g}_{j j} = = \frac{{y the y}_{22} π π}{22 ((256256 - - 11))} \\ {b b}_{j j} = = \frac{{y the y}_{33} π π}{22 ((256256 - - 11))} \end{matrix}$

令make

$\{\begin{matrix} {G G}_{rgb rgb} = = \sqrt{{Σ Σ}_{i i = = 00}^{22^{n no} - - 11} (({r r}_{i i}^{22} + + {g g}_{i i}^{22} + + {b b}_{i i}^{22}))} \\ {θ θ}_{rj r j} = = \frac{{r r}_{j j}}{{G G}_{rgb rgb}},, {θ θ}_{gj gj} = = \frac{{g g}_{j j}}{{G G}_{rgb rgb}},, {θ θ}_{bj bj} = = \frac{{b b}_{j j}}{{G G}_{rgb rgb}} \end{matrix}$

将和分别代替公式(5)中的θ_j，本发明得到如下的三个NASS态Will and Replacing θ _j in formula (5) respectively, the present invention obtains the following three NASS states

$\{\begin{matrix} | | {ψ ψ}_{Ar Ar} > > = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} \sqrt{33} {θ θ}_{rj r j} | | j j > > = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} \sqrt{33} {θ θ}_{rj r j} | | {v v}_{11} > > | | {v v}_{22} > > . . . . . . | | {v v}_{k k} > > \\ | | {ψ ψ}_{Ag Ag} > > = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} \sqrt{33} {θ θ}_{gj gj} | | j j > > = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} \sqrt{33} {θ θ}_{gj gj} | | {v v}_{11} > > | | {v v}_{22} > > . . . . . . | | {v v}_{k k} > > \\ | | {ψ ψ}_{Ab Ab} > > = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} \sqrt{33} {θ θ}_{bj bj} | | j j > > = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} \sqrt{33} {θ θ}_{bj bj} | | {v v}_{11} > > | | {v v}_{22} > > . . . . . . | | {v v}_{k k} > > \end{matrix} - - - - - - ((1111))$

根据公式(11)，为了表示多维彩色图像，本发明定义According to the formula (11), in order to represent the multi-dimensional color image, the present invention defines

$\begin{matrix} | | {ψ ψ}_{C C} > > = = \frac{11}{\sqrt{33}} | | {ψ ψ}_{Ar Ar} > > | | 0101 > > + + \frac{11}{\sqrt{33}} | | {ψ ψ}_{Ag Ag} > > | | 1010 > > + + \frac{11}{\sqrt{33}} | | {ψ ψ}_{Ab Ab} > > | | 1111 > > \\ = = {Σ Σ}_{j j = = 00}^{22^{n no} - - 11} (({θ θ}_{rj r j} | | j j > > | | 0101 > > + + {θ θ}_{gj gj} | | j j > > | | 1010 > > + + {θ θ}_{bj bj} | | j j > > | | 1111 > >)) \end{matrix} - - - - - - ((1212))$

其中j＝j₁…j_n整数j的二进制展开，|j>是n量子比特张量积|j₁…j_n>的简化表示，表示一副多维图像的像素的坐标。θ_rj，θ_gj和θ_bj分别表示在坐标|j>处的像素的红色灰度分量、绿色灰度分量和蓝色灰度分量。Where j=j ₁ ... j _n is the binary expansion of the integer j, and |j> is a simplified representation of the n-qubit tensor product |j ₁ ... j _n >, representing the coordinates of the pixels of a multi-dimensional image. θ _rj , θ _gj and θ _bj represent the red grayscale component, green grayscale component and blue grayscale component of the pixel at the coordinate |j>, respectively.

由于because

$| | | | | | {ψ ψ}_{C C} > > | | | | = = \sqrt{{Σ Σ}_{i i = = 00}^{22^{n no} - - 11} (({θ θ}_{ri the ri}^{22} + + {θ θ}_{gi gi}^{22} + + {θ θ}_{bi bi}^{22}))} = = 11$

因此本发明将|ψ_C>称为NASSTC。Therefore, the present invention refers to |ψ _C > as NASSTC.

(2)基于NASSTC的多维量子彩色图像的存储实现(2) Storage realization of multi-dimensional quantum color image based on NASSTC

为了方便应用图7中的NASS实现线路，本发明将这个线路命名为RNASS(Realization of NASS)线路，RNASS的符号表示如图8所示。In order to facilitate the implementation of the NASS circuit in FIG. 7 , the present invention names this circuit as RNASS (Realization of NASS) circuit, and the symbol of RNASS is shown in FIG. 8 .

NASSTC态|ψ_C>的实现线路如图9所示。The realization circuit of NASSTC state |ψ _C > is shown in Fig.9.

这样，可以三个NASS的实现线路和一些受控非门实现NASSTC态，实现了基于NASSTC的多维量子彩色图像的存储。In this way, the NASSTC state can be realized by three NASS implementation circuits and some controlled NOT gates, and the storage of multi-dimensional quantum color images based on NASSTC is realized.

本发明的有益效果是：本发明一方面解决了量子图像处理的基础问题“如何将图像储存在量子系统”，为后续的量子图像处理提供了必要的基础。本发明体现了量子图像处理在信息存储方面优于传统图像处理，如果把NASS看作是一种压缩编码方式，那么它的压缩能力是传统压缩编码无法比拟的。例如，假设用一个NASS量子态表示一幅1024×1024×1024的三维灰度图像，并假设随着量子技术的发展，实现1量子比特的代价与实现1比特的代价相当，这时，图像的压缩比为1024×1024×1024/30≈35791394。而NASSTC是在NASS的基础上，专门为存储RGB彩色图像而设计的。The beneficial effects of the present invention are: on the one hand, the present invention solves the basic problem of quantum image processing "how to store images in a quantum system", and provides a necessary basis for subsequent quantum image processing. The invention shows that quantum image processing is superior to traditional image processing in terms of information storage. If NASS is regarded as a compression coding method, its compression ability is incomparable to that of traditional compression coding. For example, assuming that a NASS quantum state is used to represent a 1024×1024×1024 three-dimensional grayscale image, and assuming that with the development of quantum technology, the cost of realizing 1 qubit is equivalent to the cost of realizing 1 bit, then the image’s The compression ratio is 1024×1024×1024/30≈35791394. On the basis of NASS, NASSTC is specially designed for storing RGB color images.

本发明能用在需要多维图像压缩的很多实际的图像处理应用领域，并对量子计算理论完善和应用的推广有重大意义。The invention can be used in many practical image processing application fields that require multi-dimensional image compression, and has great significance for the improvement of quantum computing theory and the promotion of applications.

附图说明Description of drawings

图1单量子比特U门的符号表示；Fig. 1 Symbolic representation of a single-qubit U-gate;

图2常用的单量子比特门；Figure 2 Commonly used single-qubit gates;

图3双量子比特受控U门；Figure 3 Double-qubit controlled U-gate;

图4n量子比特受控门；Figure 4n qubit controlled gate;

图5一个量子态表示的4×4×2的3维灰度图像例子；Figure 5 is an example of a 4×4×2 3-dimensional grayscale image represented by a quantum state;

图6受控R_xji算子的线路实现；The circuit realization of Fig. 6 controlled R _xji operator;

图7基于NASS多维量子灰度图像的存储实现；Figure 7 is based on the storage implementation of NASS multi-dimensional quantum grayscale images;

图8RNASS线路的符号表示；Figure 8 Symbolic representation of RNASS lines;

图9基于NASSTC多维量子彩色图像的存储实现；Figure 9 is based on the storage implementation of NASSTC multi-dimensional quantum color images;

图10虚框i＝i₁i₂的实现线路。The realization circuit of virtual box i=i ₁ i ₂ in Fig.10.

具体实施方式Detailed ways

本发明中基于NASS的多维量子灰度图像的存储实现如图7所示。|0>ⁱ表示第i个量子比特输入，|0>¹|0>²…|0>^n-1|0>ⁿ是n量子比特输入，也可以表示成第j个虚框中的量子线路实现公式(10)中的酉算子R_xj，j＝1,2,…,n。量子线路的执行顺序是从左到右，即，按虚框1,2,…,n的顺序执行。The storage implementation of the NASS-based multi-dimensional quantum grayscale image in the present invention is shown in FIG. 7 . |0> ⁱ represents the i-th qubit input, |0> ¹ |0> ² …|0> ^n-1 |0> ⁿ is the input of n qubits, which can also be expressed as The quantum circuit in the jth imaginary box realizes the unitary operator R _xj in formula (10), j=1,2,...,n. The execution order of the quantum circuits is from left to right, that is, they are executed in the order of virtual boxes 1, 2, ..., n.

本发明中基于NASSTC的多维量子彩色图像的存储实现如图9所示。将图9中的虚框1中的线路应用在初态上，得到The storage implementation of the multi-dimensional quantum color image based on NASSTC in the present invention is shown in FIG. 9 . Apply the circuit in the dashed box 1 in Figure 9 to the initial state up, get

${| | {ψ ψ}_{C C} > >}^{11} = = \frac{11}{\sqrt{33}} {| | 00 > >}^{&CircleTimes; &CircleTimes; n no} ((| | 0101 > > + + | | 1010 > > + + | | 1111 > >))$

将图9中的虚框2中的线路应用在|ψ_C>¹上，得到Applying the line in dashed box 2 in Fig. 9 to |ψ _C > ¹ , we get

${| | {ψ ψ}_{C C} > >}^{22} = = \frac{11}{\sqrt{33}} | | {ψ ψ}_{Ar Ar} > > | | 0101 > > + + \frac{11}{\sqrt{33}} {| | 00 > >}^{&CircleTimes; &CircleTimes; n no} ((| | 1010 > > + + | | 1111 > >))$

将图9中的虚框3中的线路应用在|ψ_C>²上，得到Applying the line in dashed box 3 in Fig. 9 to |ψ _C > ² , we get

${| | {ψ ψ}_{C C} > >}^{33} = = \frac{11}{\sqrt{33}} | | {ψ ψ}_{Ar Ar} > > | | 0101 > > + + \frac{11}{\sqrt{33}} | | {ψ ψ}_{Ag Ag} > > | | 0101 > > + + \frac{11}{\sqrt{33}} {| | 00 > >}^{&CircleTimes; &CircleTimes; n no} | | 1111 > >$

将图9中的虚框4中的线路应用在|ψ_C>³上，得到公式(12)中的|ψ_C>，从而实现基于NASSTC的多维量子彩色图像的存储。Apply the circuit in virtual box 4 in Figure 9 to |ψ _C > ³ to get |ψ _C > in formula (12), so as to realize the storage of multi-dimensional quantum color images based on NASSTC.

图9中的虚框i＝i₁i₂(i₁,i₂∈{0,1})的实现线路见图10。例如，当i₁＝0，i₂＝1时，图10中的线路就是图9中的虚框2的线路。See FIG. 10 for the implementation circuit of the imaginary box i=i ₁ i ₂ (i ₁ , i ₂ ∈{0,1}) in FIG. 9 . For example, when i ₁ =0 and i ₂ =1, the circuit in FIG. 10 is the circuit in the dashed box 2 in FIG. 9 .

Claims

1. the design Storage of multidimensional quantum gray scale and coloured image and an implementation method, is characterized in that, described method adopts the controlled door of n quantum bit C ⁿ(U) realize standard superposition state (NASS) quantum state arbitrarily, thereby realize the storage of multidimensional quantum gray level image; Described method is realizing on the basis of NASS quantum state, the characteristic of utilizing 1 RGB colour to represent with three gray-scale values, the circuit of realizing of three NASS quantum states of Integrated using is realized three-component standard superposition state (NASSTC) quantum state arbitrarily, thereby realizes the storage of multidimensional quantum coloured image;

The Realization of Storing of described multidimensional quantum gray level image be take quantum wire model as basis, the multidimensional quantum gray level image Realization of Storing of design based on NASS;

The Realization of Storing of described multidimensional quantum coloured image be take quantum wire model as basis, the Realization of Storing of the multidimensional quantum coloured image of design based on NASSTC.

2. design Storage and the implementation method of multidimensional quantum gray scale according to claim 1 and coloured image, is characterized in that, described method makes full use of the storage capacity of quantum superposition state, can be only with one 2 of n quantum bit storage ⁿthe multidimensional gray level image of individual pixel.

3. design Storage and the implementation method of multidimensional quantum gray scale according to claim 1 and coloured image, is characterized in that, described method can be only with one 2 of n+2 quantum bit storage ⁿthe multidimensional coloured image of individual pixel.

4. design Storage and the implementation method of multidimensional quantum gray scale according to claim 1 and coloured image, is characterized in that, described multidimensional quantum gray level image Realization of Storing is:

Defining a unitary operator is

R_{x} (θ) = [\begin{matrix} \cos θ & \sin θ \\ \sin θ & - \cos θ \end{matrix}]

θ ∈ [0,2 π] wherein;

An angle sequence is presented:

\{\begin{matrix} α_{j} = \arctan \sqrt{\frac{Σ_{i_{2} . . . i_{n}} {| θ_{1 i_{2} . . . i_{n}} |}^{2}}{Σ_{i_{2} . . . i_{n}} {| θ_{0 i_{2} . . . i_{n}} |}^{2}}} & j = 1 \\ α_{j, i_{1} . . . i_{j - 1}} = \arctan \sqrt{\frac{Σ_{i_{j + 1} . . . i_{n}} {| θ_{i_{1} . . . i_{j - 1} 1 i_{j + 1} . . . i_{n}} |}^{2}}{Σ_{i_{j + 1} . . . i_{n}} {| θ_{i_{1} . . . i_{j - 1} 0 i_{j + 1} . . . i_{n}} |}^{2}}} & 2 \leq j \leq n \end{matrix}

Wherein, suppose 0i ₂i _n, 1i ₂i _n, i ₁i _j-10i _j+1i _nand i ₁i _j-11i _j+1i _nrespectively the binary expansion of integer g, h, x and y, so

θ_{{0 i}_{2} . . . i_{n}} = θ_{g},

θ_{{1 i}_{2} . . . i_{n}} = θ_{h},

θ_{i_{1} . . . i_{j - 1} 0 i_{j + 1} . . . i_{n}} = θ_{x}

With

θ_{i_{1} . . . i_{j - 1} 1 i_{j + 1} . . . i_{n}} = θ_{y}

Mean respectively the quantum state of a width multidimensional gray level image in item | g>, | h>, | x> and | the coefficient of y>; i ₁i _j-1the binary expansion of integer z, z=0,1 ..., (2 ^j-1-1);

When j>=2, define controlled R _xjioperator is:

R_{xji} = (Σ_{h = 0, h &NotEqual; i}^{2^{j - 1} - 1} | h > < h |) &CircleTimes; I + | i > < i | &CircleTimes; R_{x} (α_{j, i})

0≤i≤2 wherein ^j-1, i=i ₁i _j-1the binary expansion of integer i, angle

Definition unitary operator R _xjfor:

\{\begin{matrix} R_{xj} = R_{x} (α_{1}) &CircleTimes; I^{&CircleTimes; (n - 1)} & j = 1 \\ R_{xj} = Π_{i = 0}^{2^{j - 1} - 1} (R_{xji} &CircleTimes; I^{&CircleTimes; (n - j)}) & 2 \leq j \leq n \end{matrix}

Successively by unitary operator R _xjact on initial state upper, just can obtain a NASS state:

| ψ_{k} > = (Π_{j = 1}^{n} R_{xj}) {| 0 >}^{&CircleTimes; n} = Σ_{i = 0}^{2^{n} - 1} θ_{i} | i > = Σ_{i = 0}^{2^{n} - 1} θ_{i} | v_{1} > | v_{2} > . . . | v_{k} >

Thereby realized the storage of the multidimensional quantum gray level image based on NASS.

5. design Storage and the implementation method of multidimensional quantum gray scale according to claim 1 and coloured image, is characterized in that, described multidimensional quantum coloured image Realization of Storing is:

Because a RGB colour can be regarded three greyscale color as, therefore a width multidimensional coloured image can be regarded the combination of three width multidimensional gray level images as, that is, in order to represent multidimensional coloured image, definition NASSTC state is:

| ψ_{C} > = Σ_{j = 0}^{2^{n} - 1} (θ_{rj} | j > | 01 > + θ_{gj} | j > | 10 > + θ_{bj} | j > | 11 >)

J=j wherein ₁j _nthe binary expansion of integer j, | j> is n quantum bit tensor product | j ₁j _nthe reduced representation of >, the coordinate of the pixel of expression one secondary multidimensional image; θ _rj, θ _gjand θ _bjbe illustrated respectively in coordinate | the red gray component of the pixel at j> place, green gray component and blue gray component;

Like this, can three NASS realize circuit and some controlled not-gates are realized NASSTC state, realize the storage of the multidimensional quantum coloured image based on NASSTC.