量子神经网络模型研究
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摘要
量子计算和神经网络结合而产生的量子神经网络(Quantum Neural Network,QNN)是新兴和前沿的学科之一,目前在全世界还处于研究者个体探索的阶段,发展还很不成熟。科学家研究量子神经网络一方面设计新型的量子神经网络模型,另一个方面研究某种模型的具体工作算法和实际应用。
在分析量子力学和量子计算理论中的一些原理和概念的基础上,通过设计神经网络拓扑结构和训练算法,形成新的量子神经网络模型。本论文创造性研究成果如下
(1)提出了量子M-P和感知器网络模型
利用量子线性叠加提出了量子M-P神经网络模型。并且在网络输入的量子状态为正交态和非正交态的两种情况下描述了该网络的工作原理以及它的权值更新算法。同时结合量子计算和传统的感知器网络提出了量子感知器网络模型,通过对量子感知器进行实例分析、性能分析和仿真实验表明一个单神经元量子感知器能实现单神经元经典感知器无法实现的XOR功能。
(2)提出了带权值的量子神经网络模型
在Grover量子算法的基础上,提出了一个带权值的量子神经网络和对它的训练方法,这种权值训练方法完全工作在量子机制下。这种网络能处理现实中的一些经典问题。并且基于Grover量子搜索算法的权值更新算法总能以一定的概率学习训练样本,达到网络的目的。
(3)提出了量子Hopfield神经网络模型
提出了存储矩阵元素基于概率分布的量子Hopfield神经网络模型,它的存储容量或记忆容量提高到了神经元个数的2N倍,比传统的Hopfield神经网络有了指数级的提高。并且工作过程符合量子演化过程。
(4)提出了无权值的量子神经网络模型
提出了两种无权值的量子神经网络,一种是量子竞争神经网络,它通过量子竞争能够对模式进行识别和分类。它在存储待识别的模式时存储容量或记忆容量比传统的竞争神经网络有了指数级的提高。另一种是随时间演化的量子门网络,它通过依赖于时间的初始哈密顿量所对应的本征态为量子初始态。随后哈密顿量随时间发生变化,经过一定的时间T后哈密顿量所对应的量子本征态就是网络需要的目标状态。
(5)提出了量子多模式识别网络模型
本论文为量子多模式识别网络模型设计了三种不同的量子多模式识别算法:一种是多模式高概率量子搜索算法,它通过一系列的幺正操作能在模式集中以较高的概率搜索目标,并且该算法在搜索多目标模式时能在一次算法的执行中就找到目标。另一种是带冗余项的多模式识别算法,它采用了新的模式集量子初态和量子编码设计方案。并充分利用量子计算的并行特性,可以同时对模式集中的多个模式以一定的概率进行识别。最后一种是部分多模式识别算法,它把数据库的N个搜索项分成K等份,在此基础上,它可以在数据库中以量子算法同时搜索到多个模式,并且它又比全局多模式搜索算法减少了(3b/4p)~(1/2)-π/6(b/p)~(1/2)搜索迭代次数。
Quantum neural network (QNN) is one of the young and outlying science built upon the combination of classical neural network and quantum computing. Its development just starts all over the world, which is in the state that the researcher explores it individually. The work of the scientists devoting to the QNN includes not only developing working algorithm and application of some models but also designing novel QNN models.
On the base of the analysis of some principles and concepts of the quantum mechanics and quantum computation theory, this dissertation designs topology structure or learning algorithm of neural network and then forms the new QNN models. The main contributions of this dissertation are summerized as follows:
(1)The quantum M-P and perceptron network models are proposed.
Making use of quantum linear superposition, a quantum M-P neural network is presented. Moreover, the working principle of this proposed network and its corresponding weight updating algorithm are expatiated in the two cases of input state being in the orthogonal and non-orthogonal basic set, respectively. At the same time, a monolayer quantum perceptron network is presented using some merits of quantum computation, especially quantum parallelism. The case、performance analysis and simulation on the monolayer quantum perceptron show that the proposed network with only one neuron can realize XOR function unrealizable with a classical perceptron having a neuron.
(2) A model of QNN with weight is presented.
Upon the analysis of the Grover’s quantum algorithm, a model of QNN with weight vector and its corresponding training method are proposed. It also can be shown that this model’s training method works in quantum mechanism. Results on the data set show that this network model can deal with some classical problem and the proposed weight updating algorithm based on the Grover always can learn training examples with a certain percentage.
(3) A quantum hopfield neural network (QHNN) model is proposed.
This dissertation presents a quantum Hopfield neural network (QHNN) whose elements of the storage matrix are performed in a probabilitic way. Contrasting to the conventional Hopfield neural network, the storage capacity of the QHNN is increased by a factor of 2~N, and its working process accords with quantum evolvement process.
(4) The models of QNN without weight are presented.
Two kinds of the model of QNN without weight are proposed: one is the quantum competitive neural network (QCNN) that can recognize patterns and class patterns via quantum competition. Contrasting to the conventional competitive neural network, the storage capacity or memory capacity of the QCNN is exponentially increased by a factor of 2n. Another is the time-dependent quantum gate network, which has the initial quantum state that is the eigenstate of time-dependent Hamiltonian operator. Then Hamiltonian evolve in time and the eigenstate corresponding to the final Hamiltonian is the target state of the network after the time T. Seeing from the macroscopy, this quantum target state can be considered to evolve from the initial state.
(5)A model of quantum multi-pattern recognition network is proposed.
Three kinds of quantum multi-pattern recognition algorithms are designed for the quantum multi-pattern recognition network: The first one is the multi-pattern highprobable quantum search algorithm that can search targets highprobably in the pattern sets through a series of unitary transformation. This algorithm can find goals by only one searching of the pattern. The second one is the algorithm of multi-pattern recognition with spurious items, which introduces a new design scheme of initializing quantum state and quantum encoding on the pattern set. Owing to the power of quantum parallel speciality, this method can recognize simultaneously with a certain probability multi-pattern of the pattern set. The final one is the multi-pattern partial quantum search algorithm. It separates the database of N items into K blocks and this algorithm can search concurrently multi-pattern in the database. Besides, it takes about (3b/4p)~(1/2)-π/6(b/p)~(1/2) fewer iterations than the global multi-pattern search algorithm.
在分析量子力学和量子计算理论中的一些原理和概念的基础上,通过设计神经网络拓扑结构和训练算法,形成新的量子神经网络模型。本论文创造性研究成果如下
(1)提出了量子M-P和感知器网络模型
利用量子线性叠加提出了量子M-P神经网络模型。并且在网络输入的量子状态为正交态和非正交态的两种情况下描述了该网络的工作原理以及它的权值更新算法。同时结合量子计算和传统的感知器网络提出了量子感知器网络模型,通过对量子感知器进行实例分析、性能分析和仿真实验表明一个单神经元量子感知器能实现单神经元经典感知器无法实现的XOR功能。
(2)提出了带权值的量子神经网络模型
在Grover量子算法的基础上,提出了一个带权值的量子神经网络和对它的训练方法,这种权值训练方法完全工作在量子机制下。这种网络能处理现实中的一些经典问题。并且基于Grover量子搜索算法的权值更新算法总能以一定的概率学习训练样本,达到网络的目的。
(3)提出了量子Hopfield神经网络模型
提出了存储矩阵元素基于概率分布的量子Hopfield神经网络模型,它的存储容量或记忆容量提高到了神经元个数的2N倍,比传统的Hopfield神经网络有了指数级的提高。并且工作过程符合量子演化过程。
(4)提出了无权值的量子神经网络模型
提出了两种无权值的量子神经网络,一种是量子竞争神经网络,它通过量子竞争能够对模式进行识别和分类。它在存储待识别的模式时存储容量或记忆容量比传统的竞争神经网络有了指数级的提高。另一种是随时间演化的量子门网络,它通过依赖于时间的初始哈密顿量所对应的本征态为量子初始态。随后哈密顿量随时间发生变化,经过一定的时间T后哈密顿量所对应的量子本征态就是网络需要的目标状态。
(5)提出了量子多模式识别网络模型
本论文为量子多模式识别网络模型设计了三种不同的量子多模式识别算法:一种是多模式高概率量子搜索算法,它通过一系列的幺正操作能在模式集中以较高的概率搜索目标,并且该算法在搜索多目标模式时能在一次算法的执行中就找到目标。另一种是带冗余项的多模式识别算法,它采用了新的模式集量子初态和量子编码设计方案。并充分利用量子计算的并行特性,可以同时对模式集中的多个模式以一定的概率进行识别。最后一种是部分多模式识别算法,它把数据库的N个搜索项分成K等份,在此基础上,它可以在数据库中以量子算法同时搜索到多个模式,并且它又比全局多模式搜索算法减少了(3b/4p)~(1/2)-π/6(b/p)~(1/2)搜索迭代次数。
Quantum neural network (QNN) is one of the young and outlying science built upon the combination of classical neural network and quantum computing. Its development just starts all over the world, which is in the state that the researcher explores it individually. The work of the scientists devoting to the QNN includes not only developing working algorithm and application of some models but also designing novel QNN models.
On the base of the analysis of some principles and concepts of the quantum mechanics and quantum computation theory, this dissertation designs topology structure or learning algorithm of neural network and then forms the new QNN models. The main contributions of this dissertation are summerized as follows:
(1)The quantum M-P and perceptron network models are proposed.
Making use of quantum linear superposition, a quantum M-P neural network is presented. Moreover, the working principle of this proposed network and its corresponding weight updating algorithm are expatiated in the two cases of input state being in the orthogonal and non-orthogonal basic set, respectively. At the same time, a monolayer quantum perceptron network is presented using some merits of quantum computation, especially quantum parallelism. The case、performance analysis and simulation on the monolayer quantum perceptron show that the proposed network with only one neuron can realize XOR function unrealizable with a classical perceptron having a neuron.
(2) A model of QNN with weight is presented.
Upon the analysis of the Grover’s quantum algorithm, a model of QNN with weight vector and its corresponding training method are proposed. It also can be shown that this model’s training method works in quantum mechanism. Results on the data set show that this network model can deal with some classical problem and the proposed weight updating algorithm based on the Grover always can learn training examples with a certain percentage.
(3) A quantum hopfield neural network (QHNN) model is proposed.
This dissertation presents a quantum Hopfield neural network (QHNN) whose elements of the storage matrix are performed in a probabilitic way. Contrasting to the conventional Hopfield neural network, the storage capacity of the QHNN is increased by a factor of 2~N, and its working process accords with quantum evolvement process.
(4) The models of QNN without weight are presented.
Two kinds of the model of QNN without weight are proposed: one is the quantum competitive neural network (QCNN) that can recognize patterns and class patterns via quantum competition. Contrasting to the conventional competitive neural network, the storage capacity or memory capacity of the QCNN is exponentially increased by a factor of 2n. Another is the time-dependent quantum gate network, which has the initial quantum state that is the eigenstate of time-dependent Hamiltonian operator. Then Hamiltonian evolve in time and the eigenstate corresponding to the final Hamiltonian is the target state of the network after the time T. Seeing from the macroscopy, this quantum target state can be considered to evolve from the initial state.
(5)A model of quantum multi-pattern recognition network is proposed.
Three kinds of quantum multi-pattern recognition algorithms are designed for the quantum multi-pattern recognition network: The first one is the multi-pattern highprobable quantum search algorithm that can search targets highprobably in the pattern sets through a series of unitary transformation. This algorithm can find goals by only one searching of the pattern. The second one is the algorithm of multi-pattern recognition with spurious items, which introduces a new design scheme of initializing quantum state and quantum encoding on the pattern set. Owing to the power of quantum parallel speciality, this method can recognize simultaneously with a certain probability multi-pattern of the pattern set. The final one is the multi-pattern partial quantum search algorithm. It separates the database of N items into K blocks and this algorithm can search concurrently multi-pattern in the database. Besides, it takes about (3b/4p)~(1/2)-π/6(b/p)~(1/2) fewer iterations than the global multi-pattern search algorithm.
引文
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