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1、<p>  中文2718字,1680單詞</p><p><b>  原文:</b></p><p><b>  Pager2</b></p><p>  Genetic Algorithm Based-On the Quantum Probability</p><p>  Bin LI

2、 and Zhen-quan Zhuang</p><p>  Laboratory of Quantum Communication and Quantum Computation,</p><p>  University of Science and Technology of China, Hefei, 230026, China</p><p>  Abs

3、tract: A genetic algorithm based on the quantum probability representation(GAQPR) is proposed, in which each individual evolves independently; a new crossover operator is designed to integrate searching processes of mult

4、iple individuals into a more efficient global searching process; a new mutation operator is also proposed and analyzed. Optimization capability of GAQPR is studied via experiments on function optimization, results of exp

5、eriments show that, for multi-peak optimization problem, </p><p>  1 Introduction</p><p>  Research development in quantum computation presents us not only with a tempting perspective of future

6、computational capability [1], but also with inspirations of improving classical algorithms by reconsidering them from a standpoint of quantum mechanics. Genetic algorithm is a well-known heuristic searching algorithm, an

7、d has been proved successful in many applications [2]. Research work on merging genetic algorithm and quantum computation has been started by some researchers since1990’s. Only </p><p>  QIGA (Quantum-Inspir

8、ed Genetic Algorithm), proposed by Ajit Narayanam, introduces the theory of many universes in quantum mechanics into the implementation of genetic algorithm [3]. The main contribution of [3] is that it proves the efficie

9、ncy of the strategy that uses multiple colonies to search in parallel, and uses a joint cross-over operator to enable the information exchange among colonies.</p><p>  Kuk-Hyun Han proposed a Genetic Quantum

10、 Algorithm (GQA) [4], in which the probability amplitude of qubit was used for the first time to encode the chromosome, and the formula of quantum rotation gate was used to implement the updating of chromosome. GQA is ba

11、sically a probability algorithm, not a genetic algorithm. All individuals evolve towards one Contemporary Evolutionary Target (CET). Important genetic operators, such as crossover and mutation, are not adopted in it.<

12、/p><p>  In this paper, a new Genetic Algorithm based on the Quantum Probability Representation (GAQPR) is proposed, in which each individual has its own Contemporary Evolutionary Target (CET) and evolves indep

13、endently; a new crossover operator is designed to enable the efficient exchange of evolution information between different individuals. A new mutation operator is also proposed, its effect is studied in experiments. Expe

14、riments on two typical function optimization problems prove that, for multipeak</p><p>  The remaining part of this paper is organized as follow: section 2 describes the principle and procedure of GAQPR, whi

15、ch includes the introduction of representation and updating strategy of chromosomes, main procedure of GAQPR, and procedure of the new designed crossover and mutation operators. Section 3 is the description of experiment

16、. Section 4 is the conclusion of the whole paper.</p><p><b>  2 GAQPR</b></p><p>  2.1 Representation and Updating of Chromosomes</p><p>  In quantum computation, the el

17、ementary unit for storing information is a quantum system with two states, called quantum bit (qubit). The key characteristic that makes qubit differ from classical bit is that it can be at the superposition of two quant

18、um states simultaneously. This superposition can be expressed as follow:</p><p>  |Ψ> = α| 0>+β|1>, (1) </p><p>  where ( α, β) is a pair of complex invariables, called pro

19、bability amplitude of qubit, which satisfies</p><p>  |α|2 + |β|2 = 1 (2)</p><p>  |0> and |1> represent two different states of qubit respectively, so one qubit can store t

20、he information of both states at the same time.</p><p>  In GQA, each gene has only two states, so one qubit is enough for encoding one gene. But it is often the case in applications that one gene may have m

21、ore than two states. In this paper, following the binary coding methods in classical genetic algorithm, we use more than one qubit to encode gene with more than two states. A chromosome is defined as follow:</p>&

22、lt;p>  Where qtj is the j-th chromosome in the colony at generation t, m is the number of genes in a chromosome, and k is the number of qubit used to encode one gene, which can be calculated by function</p><

23、;p>  k = ceil(log n2) , (4)</p><p>  where n is the state number of each gene, function ceil(x) finds the nearest integer from x towards +</p>

24、;<p>  Gene updating follows the formula of quantum rotation gate shown as follow:</p><p>  where (αi , β i) is the pair of probability amplitude of the ith qubit in chromosome,θi = s (αi βi)△θi [4],

25、and the values of s(αi, βi) and△θi are determined by a predefined strategy shown in table 1. The strategy proposed in [4] is designed for the knapsack problem, which has bias. In this paper we do some adjustment to make

26、it a universal strategy.</p><p>  Table 1. Tuning strategy of rotation angle</p><p>  Tuning pace of rotation angle “Delta” may take different values in different operations. For example, in reg

27、ular evolution operation “Delta” can take the value a little larger than that in crossover operation. This provides an easy way to control the effect of crossover operator. </p><p>  2.2 Procedure of GAQPR&l

28、t;/p><p>  The procedure of GAQPR is shown as follow:</p><p><b>  Begin</b></p><p><b>  1) t=0</b></p><p>  2) Initialize Q(t) = {qt1,qt2,…,q t n}&

29、lt;/p><p>  3) For each chromosome q t j, apply one measurement to get one solutions ptj , calculate the fitness of ptj, f(ptj) , store the solution{ ptj, f(ptj) } in T(t,j), which represents the CET of the j-t

30、h chromosome at next generation t+1.</p><p>  4) While ( t<Max_Gen ) do</p><p><b>  5) Begin</b></p><p><b>  a) t=t+1</b></p><p>  b) For eac

31、h chromosome ptj , apply one measurement to get one solution ptj ,evaluate ptj , to get f(ptj).</p><p>  c) Update ptj for one time according to its CET T(t, j).</p><p>  d) If f(ptj)>T(t,j)

32、(2), update T(t,j) with { ptj , f(ptj)},</p><p>  e) Otherwise, keep T(t ,j) unchanged.</p><p>  f) Perform new-designed genetic operator (crossover, mutation) on Q(t).</p><p><b

33、>  6) End</b></p><p><b>  End</b></p><p>  Step2) and 3) are the initialization of individuals and their CETs. Step5) to 6) is the evolution procedure of every generation, i

34、t can be seen that before step f) each chromosome evolves independently, step f) implements the interconnection between chromosomes by crossover operator.</p><p>  2.3 Crossover Operator </p><p>

35、;  In GAQPR, each chromosome represents a superposition of all possible solutions in certain distribution, any operation performed on such chromosome will affect all possible solutions it represents, thus the genetic ope

36、rators defined on the quantum probability representation have to satisfy the requirement that it must be of the same efficiency to all the possible solutions one chromosome represents.</p><p>  In classical

37、genetic algorithm, the purpose of crossover is to exchange information between different individuals. In GAQPR, the evolution information of each individual is well contained in its CET, the best solution so far and its

38、fitness. We designed a new crossover operator, shown as below, which satisfies the above requirement.</p><p>  1) Select two chromosomes from the group randomly with a given probability Pc,</p><p&

39、gt;  2) Exchange their evolution targets temporarily,</p><p>  3) Update the two chromosomes according to their new targets for one time,</p><p>  4) Change back their evolution targets.</p&g

40、t;<p>  Because CET represents the current evolution state of one individual, by exchanging CETs of two individuals, the evolution process of one individual will be influenced by the evolution state of the other o

41、ne. </p><p>  2.4 Mutation Operator </p><p>  The purpose of mutation is to slightly disturb the evolution states of some individuals, to prevent the algorithm from falling into local optimum. T

42、he requirement for designing mutation resembles that for designing crossover. As a probing research, we define a single qubit mutation operator, but the thought can be generalized easily to the multiple qubits scenarios.

43、 Following is the procedure of mutation operator.</p><p>  1) Select a set of chromosomes with a given probability Pm.</p><p>  2) For each chromosome, select a qubit randomly</p><p&g

44、t;  3) Exchange the position of its pair of probability amplitude,αti and βti.</p><p>  Obviously, the mutation operator defined above has the same efficiency to all the superposition states.</p><

45、p>  3 Experiments and Discussion</p><p>  Two typical function optimization problems, single-peak function optimization and multi-peak function optimization, are used to test and compare the capability of

46、 algorithms. To guarantee the objectivity of the experiment results, every curve illustrated in following figures is the mean of 20 independent experiments.</p><p>  Function (6) is a single-peak function to

47、 be optimized, which is also used as the fitness function of individuals.</p><p>  Fig. 1 shows the performances of GAQPR and GQA. It can be seen that both algorithms are efficient for the single-peak optimi

48、zation problem.</p><p>  A variant of Rosenbrock function (7) is used as a multiple-peak function to be optimized, which is also used as the fitness function of individuals.</p><p>  Fig. 2 show

49、s the performances of GAQPR and GQA. It can be seen that, for the optimization of function (7), GAQPR is much more efficient than GQA, and that GQA tends to be premature.</p><p>  To test the effect of the n

50、ew mutation operator, we introduced only the mutation operator into GQA to optimize the function (7). Fig. 3 shows the results of experiment. It can be seen that, when adopted alone, the new designed mutation operator ca

51、n help GQA jump out of local optimum efficiently.</p><p>  We studied also the effect of mutation operator in GAQPR when adopted together with the crossover operator. Fig. 4 shows the performances of GAQPR w

52、ith various probability of mutation Pm. It can be seen that, when works together with crossover operator, the effect of mutation is not so obvious, and the probability Pm can’t be large, otherwise the performance will de

53、crease. This may be explained as follow: In GAQPR, multiple individuals evolve independently towards multiple different directions s</p><p>  4 Conclusion</p><p>  Quantum probability representa

54、tion of chromosome provides the individual with much more powerful global searching capability than that of classical one. In GAQPR, these global searching capabilities of multiple individuals are exploited and integrate

55、d by adopting a new searching strategy, which utilizes multiple individuals to search in parallel and uses a new designed crossover operator to implement the communication between individuals. Experiment results prove th

56、at, for multi-peak optimizatio</p><p>  References</p><p>  1. Divincenzo D P, Quantum Computation, Science, 1995, 10, 255~261.</p><p>  2. Guo-liang Chen, Xu-fa Wang, Zhen-quan Zhu

57、ang, Dong-sheng Wang, Genetic algorithmand its applications, people’s post and telecommunications press, June 1996. (in Chinese)</p><p>  3. Narayanan, A. and Moore, M. Quantum inspired genetic algorithms. I

58、n Proceedings of the1996 IEEE International Conference on Evolutionary Computation(ICEC96). IEEE Press.1996.</p><p><b>  譯文: </b></p><p>  基于量子概率的遺傳算法</p><p><b>

59、;  李斌和莊振全</b></p><p>  量子通信和量子計(jì)算的實(shí)驗(yàn)室,</p><p>  中國(guó)科學(xué)技術(shù)大學(xué),中國(guó),合肥</p><p>  摘要:基于量子概率表示的遺傳算法的(GAQPR)提出,它是每個(gè)個(gè)體獨(dú)立的進(jìn)化;一種新的交叉操作的目的是將集體搜索多個(gè)個(gè)體的過程轉(zhuǎn)化為一種更加有效全局搜索過程。分析提出了一個(gè)新的變異算子。GAQPR的優(yōu)化能力通

60、過函數(shù)優(yōu)化實(shí)驗(yàn)進(jìn)行研究,實(shí)驗(yàn)結(jié)果表明,對(duì)于多峰優(yōu)化問題,GAQPR比GQA更高效。</p><p><b>  介紹</b></p><p>  研究量子計(jì)算的發(fā)展展現(xiàn)給我的不僅是誘人的未知計(jì)算能力[1],但也給了我們一種靈感從量子力學(xué)計(jì)算的的角度去改進(jìn)傳統(tǒng)的遺傳算法。遺傳算法是一個(gè)著名的探索式搜索算法,并已證明在許多應(yīng)用中是成功的[2]。從1990年開始一些研究者已

61、經(jīng)開始結(jié)合遺傳算法和量子計(jì)算進(jìn)行研究。迄今為止只提出了兩個(gè)實(shí)用模型。</p><p>  QIGA(量子激發(fā)遺傳算法),是Ajit Narayanam提出的,它介紹了將許多宇宙量子力學(xué)思想應(yīng)用到遺傳算法[3]。主要貢獻(xiàn)在于,它證明了使用多個(gè)種群集合并行搜索,并使用一個(gè)聯(lián)合交叉算子使種群集合之間的進(jìn)行信息交換的策略是有效的。</p><p>  Kuk-Hyun Han提出了遺傳量子算法(G

62、QA)[4],它首次將概率振幅量子位用于染色體編碼,量子旋轉(zhuǎn)門的公式是用來實(shí)現(xiàn)染色體的更新。GQA基本上是一個(gè)概率算法,不是一個(gè)遺傳算法。所有個(gè)體的進(jìn)化都朝著一個(gè)進(jìn)化目標(biāo)(CET)。重要的遺傳操作、像交叉和變異等都不采用。</p><p>  在本文中,提出了一種新的基于量子概率表示的遺傳算法 (GAQPR),在這個(gè)算法里每個(gè)個(gè)體都有屬于自己的現(xiàn)在進(jìn)化的目標(biāo)(CET)和獨(dú)立的進(jìn)化,一種新的交叉算子的設(shè)計(jì)能夠使不同

63、的個(gè)體之間進(jìn)化高效的信息交換。也提出一個(gè)新的變異算子,研究實(shí)驗(yàn)表明它是有效的。實(shí)驗(yàn)證明兩個(gè)典型的函數(shù)優(yōu)化問題,對(duì)于多峰型優(yōu)化問題,GAQPR比GQA更有效率。</p><p>  本文的其余部分內(nèi)容如下:第二部分描述GAQPR的原理和過程,包括染色體的表示和更新策略的引入,GAQPR主要的程序,設(shè)計(jì)的新的交叉和變異操作程序。第三節(jié)描述的實(shí)驗(yàn)。第四部分是整個(gè)論文的總結(jié)。</p><p>&l

64、t;b>  GAQPR</b></p><p>  2.1染色體的更新與表示</p><p>  在量子計(jì)算中,存儲(chǔ)信息的基本單位是一個(gè)擁有兩個(gè)態(tài)的量子系統(tǒng),它稱為量子比特(量子位)。單量子比特不同于經(jīng)典位的關(guān)鍵特征是它可以同時(shí)在兩個(gè)量子態(tài)的疊加。這種疊加可以表示為: </p><p>  |Ψ> = α| 0>+β|1>,

65、 (1)</p><p>  (α, β)是一對(duì)不變的復(fù)數(shù),稱為量子態(tài)的概率幅,它滿足</p><p>  |α|2 + |β|2 = 1 (2)</p><p>  | 0 >和| 1 >分別代表兩種不同狀態(tài)的量子位,所以一個(gè)量子位可以同時(shí)存儲(chǔ)兩個(gè)態(tài)的信息。</p><p>

66、  在GQA中,每個(gè)基因只有兩種狀態(tài),所以一個(gè)量子位對(duì)一個(gè)基因編碼也是足夠的。但一個(gè)基因能夠擁有兩個(gè)狀態(tài)以上的程序也是時(shí)常出現(xiàn)的。在本文中,在經(jīng)典遺傳算法的二進(jìn)制編碼方法之后,我們使用不止一個(gè)單量子比特去編碼擁有超過兩個(gè)態(tài)的基因。染色體的定義是:</p><p>  qtj是第j染色在種群的第t代,m是基因在染色體的數(shù)量,和k是用于編碼一個(gè)基因需要單量子比特的數(shù)量,我們可以通過下面計(jì)算公式得到</p>

67、<p>  k=ceil(logn2) (4)</p><p>  這里n是每個(gè)基因的狀態(tài)數(shù)量,函數(shù)ceil(x)找出從x趨向于無窮的最近整數(shù)。</p><p>  基因更新遵循的量子旋轉(zhuǎn)門的公式如下所示: </p><p>  這里(αi,βi)是染色

68、體的第i個(gè)單量子比特的一對(duì)概率幅,θi = s(αiβi)△θi[4],s(αiβi)和△θi的值取決于一個(gè)預(yù)先決定好的策略,如表1所示。這個(gè)策略主要是專為背包問題[4],偏見。在本文中,我們做一些調(diào)整,使它具有普遍性。</p><p>  表1旋轉(zhuǎn)角度的優(yōu)化策略</p><p>  緩慢的調(diào)整旋轉(zhuǎn)角在不同的操作中能夠取得不同的值。例如,在普通的旋轉(zhuǎn)角進(jìn)化過程中,Delta比交叉操作取得的

69、值更大點(diǎn)。這提供了一種簡(jiǎn)單的方法來控制交叉算子的效果。</p><p>  2.2 GAQPR程序</p><p>  GAQPR的過程如下所示:</p><p><b>  開始</b></p><p><b>  1) t = 0</b></p><p>  2) 初始化

70、 Q(t)= { qt1 qt2,…,qt n }</p><p>  3) 對(duì)于每個(gè)染色體q t j ,使用一個(gè)測(cè)量方法來得到解決問題的方案ptj,計(jì)算ptj的適應(yīng)度,f(ptj),在T(t, j)里記錄解決方案{ ptj、f(ptj)},它代表了第j染色體在下一代t+1中的進(jìn)化目標(biāo)。 </p><p>  4) 當(dāng)t小于最大遺傳代數(shù)</p><p><b&

71、gt;  5) 開始</b></p><p>  a) t = t + 1</p><p>  b) 對(duì)于每個(gè)染色體q t j ,使用一個(gè)測(cè)量方法來得到解決問題的方案ptj,評(píng)估ptj,得到f(pt j)</p><p>  c) 根據(jù)其計(jì)劃目標(biāo)T(T,j)更新qt j。</p><p>  d) 如果f(ptj)> T(T

72、,j)(2),更新T(T,j)中的{ ptj,f(ptj)},</p><p>  e) 否則,T(T,j)保持不變。</p><p>  f) 對(duì)種群Q(t)執(zhí)行新設(shè)計(jì)遺傳操作(交叉,變異)。</p><p><b>  6) 結(jié)束</b></p><p><b>  結(jié)束</b></p&g

73、t;<p>  步驟2)和3)是對(duì)個(gè)體和現(xiàn)在的進(jìn)化目標(biāo)進(jìn)行初始化。步驟5)到6)是每一代的進(jìn)化過程,可以看出步驟f)之前每個(gè)染色體獨(dú)立進(jìn)化,步驟f)步驟通過交叉操作實(shí)現(xiàn)了兩個(gè)染色體之間的信息交換。</p><p><b>  2.3 交叉操作</b></p><p>  在GAQPR中,每個(gè)染色體代表在某些分布所有可能的解決方案的疊加,在這樣的染色體上執(zhí)

74、行任何操作將影響它代表所有可能的解決方案,因此, 定義在量子概率表示的遺傳算子必須滿足的要求是一個(gè)染色體代表的所有解決方案必須有同樣的效果。</p><p>  在經(jīng)典遺傳算法,交叉的目的是在不同的個(gè)體之間進(jìn)行交換信息。在GAQPR中,每個(gè)個(gè)體的進(jìn)化信息包含在他的CET中,它里面有目前最好的解決方案及其適應(yīng)度。我們?cè)O(shè)計(jì)了一種新的交叉算子,如下所示,它滿足上述要求。</p><p>  1)

75、用電腦給的概率從染色體組中選擇兩個(gè)染色體,</p><p>  2)暫時(shí)交換進(jìn)化目標(biāo),</p><p>  3) 根據(jù)他們的新目標(biāo)更新兩個(gè)染色體,</p><p>  4)恢復(fù)到他們?cè)瓉磉M(jìn)化的目標(biāo)。</p><p>  因?yàn)镃ET代表了一個(gè)個(gè)體當(dāng)前進(jìn)化狀態(tài),通過交換兩個(gè)個(gè)體的CET,一個(gè)個(gè)體的進(jìn)化過程將影響另外一個(gè)的進(jìn)化過程。</p&g

76、t;<p><b>  2.4 變異操作</b></p><p>  突變的目的是稍微亂某些個(gè)體的進(jìn)化狀態(tài),防止算法陷入局部最優(yōu)。設(shè)計(jì)突變的要求與設(shè)計(jì)交叉的相似。作為一個(gè)研究,我們定義一個(gè)簡(jiǎn)單的單量子比特變異操作,但思想可以很容易地推廣到多個(gè)單量子比特中。下面是變異算子的過程。</p><p>  1)用電腦給定的概率選擇一組染色體。</p>

77、<p>  2)對(duì)于每一個(gè)染色體,隨機(jī)選擇一個(gè)量子位</p><p>  3)交換這個(gè)位置的一對(duì)概率幅αti 和βti</p><p>  很明顯,上面定義的變異算子對(duì)所有的疊加狀態(tài)具有相同的效率。</p><p><b>  實(shí)驗(yàn)分析</b></p><p>  兩個(gè)典型的函數(shù)優(yōu)化問題,單峰函數(shù)優(yōu)化和多峰

78、函數(shù)優(yōu)化,它們用于測(cè)試和比較算法的性能。保證實(shí)驗(yàn)結(jié)果的客觀性,每個(gè)曲線所示數(shù)據(jù)是來自于20個(gè)獨(dú)立的實(shí)驗(yàn)。</p><p>  函數(shù)(6)是一個(gè)單峰函數(shù)優(yōu)化,這也是作為個(gè)體的適應(yīng)度函數(shù)。</p><p>  圖1顯示了GAQPR和GQA的性能。可以看出兩種算法對(duì)單峰優(yōu)化問題是有效的。</p><p>  一個(gè)復(fù)雜的Rosenbrock函數(shù)(7)作為多峰函數(shù)來優(yōu)化,這也

79、是作為個(gè)體的適應(yīng)度函數(shù)。</p><p>  圖2顯示了GAQPR和GQA的性能。可以看出,對(duì)于優(yōu)化函數(shù)(7), GAQPR比GQA更有效,GQA往往是不成熟的。</p><p>  為了測(cè)試新的變異算子的效果,我們只將變異算子引入GQA來優(yōu)化函數(shù)(7)。圖3顯示了實(shí)驗(yàn)的結(jié)果。可以看出,當(dāng)單獨(dú)采用新設(shè)計(jì)的變異算子可以幫助GQA有效跳出局部最優(yōu)。</p><p>  

80、我們還研究了在GAQPR采用的交叉算子時(shí)的變異算子的作用。圖4顯示了GAQPR使用各種突變概率Pm的性能。可以看出,當(dāng)和交叉算子一起使用的時(shí)候效果并不是那么明顯,和概率點(diǎn)Pm不能太大,否則降低會(huì)性能。這可以解釋為如下:在GAQPR中,許多個(gè)體朝著許多不同的目標(biāo)同時(shí)獨(dú)立的進(jìn)化。這是能夠有效地防止算法過早收斂。當(dāng)要搜索最好的解決方案的時(shí)候。太多的變異操作將導(dǎo)致算法在最佳點(diǎn)附近搖擺,然后放慢收斂的速度。</p><p>

81、;<b>  總結(jié)</b></p><p>  染色體量子概率的表示提供比經(jīng)典的全局搜索能力更加強(qiáng)大的全局搜索能力的個(gè)體。在GAQPR中,這些多個(gè)個(gè)體的全局搜索能力通過采用新的搜索策略被綜合運(yùn)用,它利用多個(gè)個(gè)體并行搜索和使用新的設(shè)計(jì)交叉算子實(shí)現(xiàn)個(gè)體之間的溝通。實(shí)驗(yàn)結(jié)果證明,對(duì)于多峰優(yōu)化問題,這種策略比GQA更為有效。</p><p><b>  引用<

82、/b></p><p>  [1]Divincenzo D P,量子計(jì)算、科學(xué),1995年,255 ~ 261。</p><p>  [2]陳國(guó)良 王旭發(fā) 莊振全 王東申,遺傳算法應(yīng)用 人民郵電出版社,1996年6月。</p><p>  [3]Narayanan, A. and Moore, M .量子遺傳算法。1996 IEEE進(jìn)化計(jì)算國(guó)際會(huì)議(ICEC9

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