It is suggested that a type of artificial neural network can be built using the principles of quantum information processing. A new technique for the recognition of noisy English letter using    quanturn neural network is proposed in this paper. A qubit neuron model based on quantum mechanics, constructed the quantum back propagation learning rule network (QBP) is given, and a high reliability has been achieved. The numerical experiments of the identification of noisy English letters based on the quantum network has been performed, it is shown that QBP can identify the noised English letters efficaciously; the recognition rate may reach the 100%when the variance of the noise is less than 0.35, and has more improvement performance in anti-noise than that of conventional back propagation neural network，,the error recognition rate decreases from 20.46% to3% when the noise variance is set to 0.40.

1.Introduction

 Artificial neural networks have been intensively studied since they have attractive characteristic, such as, in applications of pattern recognition [1], However, it faces problems in practice since the massively parallel characteristics of most models are not suitable for the simulation in a reasonable time on classical computers.
Quantum computing, that operates according to the principles of quantum mechanics, has great parallelism based on quantum states superposition. In 1994, Shor proposed a way to factorize large integers in polynomial time by using a quantum computing algorithm[2]. In 1996, Grover proposed the quantum search algorithm with only O(N ) complexity to solve the unsorted database problem with N records[3]. Integrating the characteristics of artificial neural
networks with quantum computing, quantum neural network is a promising field in information processing .
   Kak [4] and Perus [5] drew attention to the similarity between the description of neuron states and that of quantum mechanical states and discussed a notion of quantum neural computing in accordance with quantum theory. Similar approaches have also been explored by Behrman [6], Menneer[7], and Ventura [8]. In Ref.[9], Koudu proposed a quantum bit
(qubit) neuron model, which is inspired by quantum physics and quantum computing, and controlled by three parameters, that is, phase parameters 􀀭k , threshold􀁏 , and the reversal parameter􀁇 . At the same time, quantum neural network (QNN) has been studied on its application in the recognition of handwritten numerals [10].
   In this paper, a neuron model based on qubit and quantum gate is given, and only two parameters, phase shifting coefficient and phase controlled factor are designed to adjust the neuron. Besides, a three-layered quantum neural network with back propagation learning rule is constructed to recognize the noisy English letter. In Section 2, we give the qubit model and
通过量子逆传播网络
识别那些吵闹的英文字母

摘要

量子信息处理原则被建议能用于人工神经网络的建立。 一种新的使用quanturn神经网络技术用来识别那些吵闹的英文字母的技术已经在纸面上被提出了。一个量子位元神经元模型根据量子力学，构建量子反向传播来学习规则网络(QBP)已经被给出，而且其可靠性已经很高。基于量子网络的鉴定吵杂的英文字母的数值实验已经操作成功，结果表明:QBP可以识别有效采集英文字母。当噪声的方差小于0.35的时候，其识别率可以达到100%。而且，比起普通反向传播神经网络，它在抗干扰性方面有更多的改进。但当方差的噪声被设置成0.4的时候，它的错误识别率就会从20.46%降到3%。

1.介绍

因为非常得吸引人，人工神经网络已经被广泛研究，比如在识别模式[1]的应用中。但是，它在实践中面临着很多问题，因为大多数模型的大规模并行特点在合理时间都不适用于对传统电脑的仿真。
根据量子力学的原则来操作的量子计算，在量子态叠加的基础上有很大的平行度。1994年，缺少提出一种用量子计算算法[2]来在多项式时间内因式分解大量的整数的方法。1996年，格罗夫根据数据库记录[3]，仅仅利用O（N）提出了量子搜索算法来解决未分类的数据库中的备案问题。用量子计算结合人工神经网络的特点可知：量子神经网络在信息加工领域是很有前途的。
Kak[4]和Perus[5]在对神经元州的描述和量子力学状态的相似性吸引了人们的注意力，继而大家根据量子理论来讨论一种量子神经计算的想法。相似的方法也被贝尔曼[6]，曼尼尔[7]，文托拉[8]所开发。在文献[9]中，库度提出了一个量子比特（量子位元）神经元的模型，这是受量子物理和量子算法所启发，而且，由三个参数控制，它们分别是 ，相参数，阈值和逆转参数。与此同时，人们已经在研究量子神经网络（QNN）在对手写数字的识别的方面的应用。在本文中，神经元模型根据量子位元和量子门被提出了，只有两个参数，移相系数和相位控制因素是为了调整神经元。除此之外，带有反向学习规则的三层量子神经网络被建立来识别喧闹的英文信。在第二部分，我们给出了量子位元模型和他的神经网络。

its neural network. This is followed in Section 3 by numerical simulations to examine the performance of the qubit neural network. The recognition performance of the noisy English letter with quantum back propagation network is discussed and compared with that of conventional back propagation network.

2.Quantum qubit and quantum gate

A quantum state is a deterministic entity that contains all information that can be possibly learned from a quantum system. The most common representation of a quantum system is a unit column vector | >, called a ket, in a Hilbert space. It is a deterministic entity that contains all information that can be possibly learned from the quantum system. A bra <| is the transpose of a ket. A qubit is a unit vector in a two dimensional vector space for which a
particular basis, denoted by {|0>, |1>}, has been fixed, where |0> and |1> are taken to encode the classical bit values 0 and 1 respectively. Unlike classical bits, qubits can be in a superposition of |0> and |1>, such as a|0>+b|1> where a and b are complex numbers such
that |a|2+|b|2=1. A quantum state may also be expressed by its phase, such as,
          f ()=e = cos + i sin                              （1）
where i is the imaginary unit of -1 and  is the phase. Two frequently used quantum logic gates are the 1-bit rotation gate and the 2-bit controlled NOT gate. The 1-bit rotation gate rotates a quantum state input to it over the angle in the complex plane, and the controlled NOT gate performs an XOR operation to the input quantum state, that is, the output is |0> if input quantum state is |1>. Since the phase rotation gate is a phase shifting gate that transforms the phase of quantum states, it may be represented as following with Eq.(1) formalism
          f (1 +2 ) = f (1 )· f (2 )                                     （2）
And the controlled NOT gate may also be defined as
                           sin+icos      (「=1)
          f (π∕2「－ )=｛cos+i sin     (「=0 )                         (3)

where the output of gate is controlled by the input parameter 「. In the case of 「=0, the phase of the probability amplitude of quantum state |1> is reserved, however, its observed probability is invariant, therefore, it is regarded as non-rotation.

3. Quantum neuron model and quantum back propagation network

Let assume that the firing neuron state is defined as qubit state |1>, and non-firing neuron is defined as quantum state |0>, therefore, the arbitrary neuron state is the coherent superposition of the |0> and |1>. Figure 1 illustrates the quantum neuron’s convergence performance against the iteration step numbers when  is set up to 0.31. It is shown that
quatnum neuron has a good performance than that of conventional neuron with the same condition.

这在第3部分中继续,即通过数值模拟的方法来检验神经网络的量子位元的性能。人们讨论了关于识别带有量子反射传播网络的喧闹的英文字母的性能，并且与普通反向传播网络相对比。

2. 量子量子位元和量子门

一个量子态是一个确定性的实体，其中包含所有信息，它们有可能是从一个量子系统中学习而来。一个量子系统最常见的表现是一个单位列向量|􀁍>，称为为偈人,是在希耳伯特空间。这是一个确定性的实体,它包含了所有可能从量子系统学习来的信息。一个量子位元是一个单位矢量在一套空间向量空间为其中的一个特定的基础上，通过引入{ | | 0 >,1 > },已经被修正, 在那里，|0>和 |1>分别被用来编码古典位值0和1。不同于古典位元，量子位元可以叠加在|0>和|1>，例如，a|0>+b|1>中a和 b是复杂的编号，就像|a|2+|b|2=1。一个量子态也可以通过它的阶段来表示，例如：
     f ()= e = cos + i sin                                          (1)
其中i是-1里虚构的单位，是阶段。两个常用的量子逻辑门是1-bit旋转的门和2-bit控制门。在复杂的飞机上的一个角度，1-bit旋转门把一个量子状态输入其中，控制门执行XOR操作输入量子状态，那就是，若输入量子态| 1 >则输出| 0 >。由于相位旋转闸门时一种转变量子态阶段的移相，它可以用下式表示：
     f (1 +2 ) = f (1 )· f (2 )                                          （2）
 
同时，控制门也可以定义为：
   
          sin+icos      (「=1)
     f (π∕2「－ )=｛cos+i sin     (「=0 )                         (3)

其中输出门由输入参数「决定。如果「=0，量子状态的概率振幅的相位就被保留了，但是，它的观察概率在变化，因此，它被看作是non-rotation.。

3. 量子神经模型和量子反向传播网络

   假定射击神经元状态定义为普通| 1 >形式，non-firing神经元的定义为量子态| 0》，因此，任意神经元状态是叠加| | 0 >和1 >。图1阐述了当被设置为0.31的时候量子神经元的收敛域与迭代步数字的关系。可以看出，在相同条件下，量子神经元比普通神经元表现得更好。

     Figure 1 The performance of the quantum neuron
convergence rate against iteration steps when  is set to 0.31

    Figure2The illustration of three layer quantum neural
netwok with complex BP learning rule

The experiments with different show that the convergence rate becomes faster when  becomes great and great. With the learning rule of conventional complex back propagation network, a three-layered network based on this neuron model can be constructed. It is
named quantum back propagation network (QBP). A QBP network with L neurons in the input layer, K neurons in the hidden layer and M neurons in the output layer can be illustrated in Figure2.

4. Recognition of noisy English letter using quantum neural network

  图1 当被设置成0.31时，量子神经的性能与迭代步收敛速度

 图2  带有复杂的BP学习规则的三层量子神经网络的说明

以不同的做实验，结果表明，该算法的收敛速度会随着变大而越来越快。随着对传统的复杂的反向传播网络的学习，基于此神经元模型的三层网络可以开工建设了。它被命名为量子反向传播网络（QBP）。一个带有L神经元在它的输入层，K神经元在它的隐层，M神经元在它的输出层的QBP网络可以在图2中看出。

4. 使用量子神经网络技术识别那些吵闹的英文字母

Noisy English letter recognition is a simple application of neural network. We now analyze the performance of quantum neural network by employing it in the noisy English letter recognition. An English character may be expressed as a 7×5 matrix in conventional BP network recognition.  
  In order to recognize the noisy English letter by our QBP network, 35 neurons in the input layer, 11 neurons in the hidden layer, and 26 neurons in the output layer are designed in the QBP network (denoted 35-11-26). As same as the conventional BP network, both the network training and character testing are included in the recognition. Experiments are conducted on the 26 noisy English letters. As convergence condition, we define it has "learned", if square error is less than "Elower" as threshold of convergence, and it has "not learned" if square error is not less than "Elower" until "Lupper" as upper limit of iteration. Moreover, one iteration is defined that all teaching patterns are inputted to network. First, we initialize the phase shifting coefficient and phase controlled factor of the input layer by some small random complexes, which are the amplitude of 35 quantum states. They are constructed as following,
∣j>=a j∣0>+1- a j∣1>∣j=1,…35                         (4)
 Then, recognition is executed on English letter with noise. The average number of the Gaussian noise is set to zero, and the variance is set from 0 to 0.5, which is changed per 0.05, and 100 trials are executed for each noise level. Recognition Rate is computed against the
noise level in Figure 3, where the learning coefficient in QBP network is set to 0.0015 and Elower =0.001. It is shown that QBP can identify the noised English letters efficaciously; the recognition rate may reach the 100% when the variance of the Gaussian noise is less than
0.35.
 
      Figure 3 The noise level dependence of recognition rate 
 

喧闹的英文字母识别是神经网络中一个简单的应用程序。我们现在通过量子神经网络在喧闹的英文字母识别中的使用分析其性能。英语的特点在传统BP神经网络识别中可能被表示为一个7×5矩阵。
   为了用QBP网络识别喧闹的英文字母，输入层的35个神经元，隐层的11个神经元，输出层的26个神经元在QBP网络中设计（表示为35-11-26）。和传统的BP神经网络一样，网络训练和性格测试都包括在识别之中。实验在26个喧闹的英文字母中进行。作为收敛条件，如果方误差不到“Elower”，作为收敛性的阈值，则我们把它定义为“已经学了”。如果方误差不是低于“Elower”到“Lupper”
作为上限限制，则我们定义为“还没学过”。另外，一种迭代被定义为所有教学模式被输入到网络。首先，我们通过一些小的随机复杂的振幅把输入层中的移相系数和相位控制因素初始化，它们是振幅为35的量子状态。它们可表示为：
    ∣j>=a j∣0>+1- a j∣1>∣j=1,…35                         (4)
然后，识别带噪声的英文字母。高斯噪声的平均数被设置为0，方差被设置为0到0.5，每0.05改变，100个实验为每个噪声水平而执行。计算的识别率和噪声水平的关系，可在图3中看出。其中QBP网络中的学习系数被设置为0.0015，Elower=0.001.这表明，QBP可以有效地识别喧闹的英文字母；当高斯噪声的方差低于0.35的时候，识别率可以到达100%。

        图3 讯号的噪声水平识别率

5. Conclusions

A kind of quantum neural network and its simple applications in pattern recognitionion are proposed in this paper. First, a quantum neuron model, which based on the phase shift gate and controlled-Not gate, is discussed, and its convergence property is analyzed. Then, the complex quantum back propagation neural network model, based on the conventional feedforward network structure, is built with these neurons. In order to analyze the performances of QBP, the application in identifying noisy English letters is presented. The error rate of quantum networks can decrease greately, such as from 20.46% to 3% when the noise variance is set up to 0.40.

5．结论