Genetic Quantum Algorithm and its Application to Combinatorial Optimization Problem

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1 Genetic Quantu Algorith and its Application to Cobinatorial Optiization Proble Kuk-Hyun Han Dept. of Electrical Engineering, KAIST, 373-, Kusong-dong Yusong-gu Taejon, , Republic of Korea Jong-Hwan Ki Dept. of Electrical Engineering, KAIST, 373-, Kusong-dong Yusong-gu Taejon, , Republic of Korea Abstract- This paper proposes a novel evolutionary coputing ethod called a genetic quantu algorith (GQA). GQA is based on the concept and principles of quantu coputing such as qubits and superposition of states. Instead of binary, nueric, or sybolic representation, by adopting qubit chroosoe as a representation GQA can represent a linear superposition of solutions due to its probabilistic representation. As genetic operators, quantu gates are eployed for the search of the best solution. Rapid convergence and good global search capability characterize the perforance of GQA. The effectiveness and the applicability of GQA are deonstrated by experiental results on the knapsack proble, which is a well-known cobinatorial optiization proble. The results show that GQA is superior to other genetic algoriths using penalty functions, repair ethods, and decoders. Introduction Many efforts on quantu coputers have progressed actively since the early 990 s because these coputers were shown to be ore powerful than classical coputers on various specialized probles. But if there is no quantu algorith that solves practical probles, quantu coputer hardware ay be useless. It could be considered as a coputer without operating syste. Although there would be signicant benet fro new quantu algoriths that could solve coputational probles faster than classical algoriths, to date, only a few quantu algoriths are known. Nevertheless, quantu coputing is attracting serious attention, since its superiority was deonstrated by a few quantu algoriths such as Shor s quantu factoring algorith [, ] and Grover s database search algorith [3, 4]. Shor s algorith nds the prie factors of an n-digit nuber in polynoial-tie, while the best-known classical factoring algoriths require tie O n 3 log(n) 3. Grover s database search algorith can nd an ite in an unsorted list of n ites in O ( p n) steps, while classical algoriths require O(n). Research on erging evolutionary coputing and quantu coputing has been started by soe researchers since late 990 s. They can be classied into two elds. One concentrates on generating new quantu algoriths using autoatic prograing techniques such as genetic prograing [5]. The absence of new quantu algoriths otivated this work. The other concentrates on quantu-inspired evolutionary coputing for a classical coputer [6], a branch of study on evolutionary coputing that is characterized by certain principles of quantu echanics such as standing waves, interference, coherence, etc. This paper offers a novel evolutionary coputing algorith called a genetic quantu algorith (GQA). GQA is characterized by principles of quantu coputing including concepts of qubits and superposition of states. GQA uses a qubit representation instead of binary, nueric, or sybolic representations. GQA can iitate parallel coputation in classical coputers. This paper is organized as follows. Section describes a novel evolutionary coputing algorith, GQA. Section 3 contains a description of the experient with GAs and GQAs for knapsack probles for coparison purpose. Section 4 suarizes and analyzes the experiental results. Concluding rearks follow in Section 5. Genetic Quantu Algorith (GQA) GQA is based on the concepts of qubits and superposition of states of quantu echanics. The sallest unit of inforation stored in a two-state quantu coputer is called a quantu bit or qubit [7]. A qubit ay be in the state, in the 0 state, or in any superposition of the two. The state of a qubit can be represented as jψi = ffj0i + ji; () where ff and are coplex nubers that specify the probability aplitudes of the corresponding states. jffj gives the probability that the qubit will be found in 0 state and jj gives the probability that the qubit will be found in the state. Noralization of the state to unity guarantees jffj + jj =: () If there is a syste of -qubits, the syste can represent states at the sae tie. However, in the act of observing a quantu state, it collapses to a single state [8].

2 . Representation It is possible to use a nuber of different representations to encode the solutions onto chroosoes in evolutionary coputation. The classical representations can be broadly classied as: binary, nueric, and sybolic [9]. GQA uses a novel representation that is based on the concept of qubits. One qubit is dened with a pair of coplex nubers, (ff; ), as» ff ; which is characterized by () and (). And an -qubits representation is dened as» ff ; (3) ff ff where jff i j + j i j =, i =; ; ;. This representation has the advantage that it is able to represent any superposition of states. If there is, for instance, a three-qubits syste with three pairs of aplitudes such as " p :0 p 0:0 p 3 # ; (4) the state of the syste can be represented as p p p j000i + p 3 j00i + p j00i + p 3 j0i: (5) The above result eans that the probabilities to represent the state j000i, j00i, j00i, and j0i are, 3,, and 3, respectively. By consequence, the three-qubits syste of (4) has four states inforation at the sae tie. Evolutionary coputing with the qubit representation has a better characteristic of diversity than classical approaches, since it can represent superposition of states. Only one qubit chroosoe such as (4) is enough to represent four states, but in classical representation at least four chroosoes, (000), (00), (00), and (0) are needed. Convergence can be also obtained with the qubit representation. As jff i j or j i j approaches to or 0, the qubit chroosoe converges to a single state and the property of diversity disappears gradually. That is, the qubit representation is able to possess the two characteristics of exploration and exploitation, siultaneously.. GQA The structure of GQA is described in the following. procedure GQA t ψ 0 initialize Q(t) ake P (t) by observing Q(t) states store the best solution aong P (t) while (not terination-condition) do t ψ t + ake P (t) by observing Q(t ) states update Q(t) using quantu gates U (t) store the best solution aong P (t) GQA is a probabilistic algorith which is siilar to a genetic algorith. GQA aintains a population of qubit chroosoes, Q(t) = fq t ; qt ; ; qt ng at generation t, where n is the size of population, and q t j is a qubit chroosoe dened as» q t j = ff t ff t t t ff t t ; (6) where is the nuber of qubits, i.e., the string length of the qubit chroosoe, and j =; ; ;n. In the step of initialize Q(t), ff t i and t i, i =; ; ;, of all q t j, j = ; ; ;n,inq(t) are initialized with p. It eans that one qubit chroosoe, q t j j t=0 represents the linear superposition of all possible states with the sae probability: jψ q 0 j i = X k= p js ki; where S k is the k-th state represented by the binary string (x x x ), where x i, i = ; ; ;, is either 0 or. The next step akes a set of binary solutions, P (t), by observing Q(t) states, where P (t) =fx t ; x t ; ; x t ng at generation t. One binary solution, x t j, j = ; ; ;n,isa binary string of the length, and is fored by selecting each bit using the probability of qubit, either jff t i j or ji tj, i =; ; ;,ofq t j. Each solution xt j is evaluated to give soe easure of its tness. The initial best solution is then selected and stored aong the binary solutions, P (t). In the while loop, one ore step, update Q(t), is included to have tter states of the qubit chroosoes. A set of binary solutions, P (t), is fored by observing Q(t ) states as with the procedure described before, and each binary solution is evaluated to give the tness value. In the next step, update Q(t), a set of qubit chroosoes Q(t) is updated by applying soe appropriate quantu gates U (t), which is fored by using the binary solutions P (t) and the stored best solution. The appropriate quantu gates can be designed in copliance with practical probles. Rotation gates, for instance, will be used for knapsack probles in the next sec- Quantu gates are reversible gates and can be represented as unitary operators acting on the qubit basis states: U y U = UU y, where U y is the heritian adjoint of U. There are several quantu gates, such as NOT gate, Controlled NOT gate, Rotation gate, Hadaard gate, etc.[7].

3 tion, such as U ( ) =» cos( ) sin( ) sin( ) cos( ) ; (7) where is a rotation angle. This step akes the qubit chroosoes converge to the tter states. The best solution aong P (t) is selected in the next step, and if the solution is tter than the stored best solution, the stored solution is changed by the new one. The binary solutions P (t) are discarded at the of the loop. It should be noted that soe genetic operators can be applied, such as utation which creates new individuals by a sall change in a single individual, and crossover which creates new individuals by cobining parts fro two or ore individuals. Mutation and crossover can ake the probability of linear superposition of states change. But as GQA has diversity caused by the qubit representation, there is no need to use the genetic operators. If the probabilities of utation and crossover are high, the perforance of GQA can be decreased notably. In GQA, the population size, i.e., the nuber of qubit chroosoes is kept the sae all the tie. This is caused by conservation of qubits based on quantu coputing. GQA with the qubit representation can have better convergence with diversity than conventional GAs which have xed 0 and inforation. 3 Experient The knapsack proble, a kind of cobinatorial optiization proble, is used to investigate the perforance of GQA. The knapsack proble can be described as selecting fro aong various ites those ites which are ost protable, given that the knapsack has liited capacity. The 0- knapsack proble is described as: given a set of ites and a knapsack, select a subset of the ites so as to axiize the prot f (x): subject to f (x) = p i x i ; w i x i» C; where x =(x x ), x i is 0 or, p i is the prot of ite i, w i is the weight of ite i, and C is the capacity of the knapsack. In this section, soe conventional GA ethods are described to experient with the 0- knapsack proble, and the detailed algorith of GQA for the knapsack proble follows. 3. Conventional GA ethods Three types of conventional algoriths are described and tested: algoriths based on penalty functions, algoriths based on repair ethods, and algoriths based on decoders [0]. In all algoriths based on penalty functions, a binary string of the length represents a chroosoe x to the proble. The prot f (x) of each string is deterined as f (x) = p i x i Pen(x); where Pen(x) is a penalty function. There are any possible strategies for assigning the penalty function [, ]. Three types of penalties are considered, such as logarithic penalty, linear penalty, and quadratic penalty: Pen (x) = log P P ( + ρ ( w ix i C)) ; Pen (x) =ρ ( P w ix i C) ; Pen 3 (x) =(ρ ( w ix i C)) ; where ρ is ax fp i =w i g. In algoriths based on repair ethods, the prot f (x) of each string is deterined as f (x) = p i x 0 i ; where x 0 is a repaired vector of the original vector x. Original chroosoes are replaced with a 5% probability in the experient. The two repair algoriths considered here differ only in selection procedure, which chooses an ite for reoval fro the knapsack: Rep (rando repair): The selection procedure selects a rando eleent fro the knapsack. Rep (greedy repair): All ites in the knapsack are sorted in the decreasing order of their prot to weight ratios. The selection procedure always chooses the last ite for deletion. A possible decoder for the knapsack proble is based on an integer representation. Each chroosoe is a vector of integers; the i-th coponent of the vector is an integer in the range fro to i +. The ordinal representation references a list L of ites; a vector is decoded by selecting appropriate ite fro the current list. The two algoriths based on decoders considered here differ only in the procedure of building a list L of ites: Dec (rando decoding): The build procedure creates a list L of ites such that the order of ites on the list corresponds to the order of ites in the input le which is rando. Dec (greedy decoding): The build procedure creates a list L of ites in the decreasing order of their prot to weight ratios. 3. GQA for the knapsack proble The algorith of GQA for the knapsack proble is based on the structure of GQA proposed and it contains a repair

4 algorith. The algorith can be written as follows: procedure GQA t ψ 0 initialize Q(t) ake P (t) by observing Q(t) states repair P (t) store the best solution b aong P (t) while (t <MAX GEN) do t ψ t + ake P (t) by observing Q(t ) states repair P (t) update Q(t) store the best solution b aong P (t) A qubit string of the length represents a linear superposition of solutions to the proble as in (6). The length of a qubit string is the sae as the nuber of ites. The i-th ite can be selected for the knapsack with probability j i j or ( jff i j ). Thus, a binary string of the length is fored fro the qubit string. For every bit in the binary string, we generate a rando nuber r fro the range [0::]; ifr>jff i j, we set the bit of the binary string. The binary string x t j, j =; ; ;n, of P (t) represents a j-th solution to the proble. For notational siplicity, x is used instead of x t j in the following. The i-th ite is selected for the knapsack iff x i =, where x i is the i-th bit of x. The binary string x is deterined as follows: procedure ake (x) i ψ 0 while (i <) do i ψ i + if rando[0; ) > jff i j then x i ψ else x i ψ 0 The repair algorith of GQA for the knapsack proble is ipleented as follows: procedure repair (x) knapsack-overlled P ψ false if w ix i >C then knapsack-overlled ψ true while (knapsack-overlled) do s(ff i i ) x i b i f (x) i ff i i ff i i ff i i f (b) > 0 < 0 =0 =0 0 0 false true false true 0:05ß + ± 0 0 false 0:0ß + ± 0 0 true 0:05ß + 0 ± false 0:005ß + 0 ± true 0:05ß + 0 ± Table : Lookup table of i, where f ( ) is the prot, s(ff i i ) is the sign of i, and b i and x i are the i-th bits of the best solution b and the binary solution x, respectively. select an i-th ite fro the knapsack x i ψ 0 if P w ix i» C then knapsack-overlled ψ false while (not knapsack-overlled) do select a j-th ite fro the knapsack x j ψ if P w ix i >C then knapsack-overlled ψ true x j ψ 0 P The prot of a binary solution x is evaluated by p ix i, and it is used to nd the best solution b after the update of q j, j =; ; ; n. A qubit chroosoe qj is updated by using the rotation gate U ( ) of (7) in this algorith. The i-th qubit value (ff i ; i ) is updated as»»» ff 0 i cos( i ) sin( i ) f = : (8) sin( i ) cos( i ) i 0 i In this knapsack proble i is given as s(ff i i ) i. The paraeters used are shown in Table. For exaple, if the condition, f (x) f (b), is satised and x i and b i are and 0, respectively, we can set the value of i as 0:05ß and s(ff i i ) as +,,or0 according to the condition of ff i i so as to increase the probability of the state ji. The value of i has an effect on the speed of convergence, but if it is too big, the solutions ay diverge or have a preature convergence to a local optiu. The sign s(ff i i ) deterines the direction of convergence to a global optiu. The lookup table can be used as a strategy for convergence. This update procedure can be described as follows: procedure update (q) i ψ 0

5 ] of CGAs GQAs ites Pen Pen Pen3 Rep Rep Dec Dec P +R GQA() GQA(0) b prots w t(sec=run) b prots w t(sec=run) b prots w t(sec=run) Table : Experiental results of the knapsack proble: the axiu nuber of generations 500, the nuber of runs 5. P +R eans the algorith ipleented by P en and Rep, and b:, :, and w: eans best, ean, and worst, respectively. t(sec=run) represents the elapsed tie per one run, and - eans that an experient did not ade in this case. while (i <) do i ψ i + deterine i with the lookup table obtain (ff 0 i ;0 i ) as: [ff 0 i 0 i ]T = U ( i )[ff i i ] T q ψ q 0 The update procedure can be ipleented in various ethods with appropriate quantu gates. It deps on a given proble. 4 Results In all experients strongly correlated sets of data were considered: w i = uniforly rando[; 0) p i = w i +5; the average knapsack capacity was used: C = w i ; and the data les were unsorted. The population size of the eight conventional genetic algoriths (CGAs) was equal to 00. Probabilities of crossover and utation were xed: 0.65 and 0.05, respectively, as in [0]. The population size of GQA() was equal to, and the population size of GQA(0) was equal to 0, this being the only difference between GQA() and GQA(0). As a perforance easure of the algorith we collected the best solution found within 500 generations over 5 runs, and we checked the elapsed tie per one run. A Pentiu-III 500MHz was used, running Visual C Table shows the experiental results of the knapsack probles with 00, 50, and 500 ites. In the case of 00 ites, GQA yielded superior results as copared to all the other CGAs. The CGA designed by using a linear penalty function and rando repair algorith outperfored all other CGAs, but is behind GQA() as well as GQA(0) in perforance. The results show that GQA perfors well in spite of sall size of population. Judging fro the results, GQA can search solutions near the optiu within a short tie as copared to CGAs. In the cases of 50 and 500 ites, the CGA that outperfors all the other CGAs was tested for coparison purpose with GQA. The experiental results again deonstrate the superiority of GQA. Figure shows the progress of the ean of best prots and the ean of average prots of population found by GQA(), GQA(0), and CGA over 5 runs for 00, 50, and 500 ites. GQA perfors better than CGA in ters of convergence rate and nal results. In the ning of the plotting of the best prots, GQA() shows a slower convergence rate than GQA(0) and CGA due to its sall population nuber. After 50 generations, GQA(0) and GQA() aintain a nearly constant convergence rate, while CGA s convergence rate reduces substantially. After 00 generations, even though convergence rate of GQA reduces, GQAs show a faster convergence rate than CGA due to its better global search ability. GQAs nal results are larger than CGA s in 000 generations. The tency of convergence rate can be shown clearly in the results of the ean of average prots of population. In the ning, convergence rates of all the algoriths increase. But CGA aintains a nearly constant prot due to its preature convergence iediately, while GQA approaches towards the neighborhood of global optia with a constant

6 convergence rate. GQAs display no preature convergence which is a coon proble of CGAs until 000 generations. The experiental results deonstrate the effectiveness and the applicability of GQA. Especially, Figure shows the excellent global search ability and the superiority of convergence ability of GQA. 5 Conclusions This paper proposed a novel evolutionary coputing algorith, GQA with a quantu representation. GQA is based on the principles of quantu coputing such as concepts of qubits and superposition of states. GQA can represent a linear superposition of states, and there is no need to include any individuals. GQA has an excellent ability of global search due to its diversity caused by the probabilistic representation, and it can approach better solutions than CGA s in a short tie. The knapsack proble, a kind of cobinatorial optiization probles, is used to discuss the perforance of GQA. It was showed that GQA s convergence and global search ability are superior to CGA s. The experiental results deonstrate the effectiveness and the applicability of GQA. [8] A. Narayanan, Quantu coputing for ners, in Proceedings of the 999 Congress on Evolutionary Coputation, pp. 3-38, Jul 999. [9] R. Hinterding, Representation, Constraint Satisfaction and the Knapsack Proble, in Proceedings of the 999 Congress on Evolutionary Coputation, pp. 86-9, Jul 999. [0] Z. Michalewicz, Genetic Algoriths + Data Structures = Evolution Progras, Springer-Verlag, 3rd, revised and exted edition, 999. [] J.-H. Ki and H. Myung, Evolutionary Prograing Techniques for Constrained Optiization Probles, IEEE Transactions on Evolutionary Coputation, Vol., No., pp. 9-40, Jul 997. [] X. Yao, Evolutionary Coputation: Theory and Applications, World Scientic, Singapore, 999. References [] P. W. Shor, Quantu Coputing, Docuenta Matheatica, vol. Extra Volue ICM, pp , 998, EMIS/ journals/ DMJDMV/ xvol-ic/ 00/ Shor.MAN.htl. [] P. W. Shor, Algoriths for Quantu Coputation: Discrete Logariths and Factoring, in Proceedings of the 35th Annual Syposiu on Foundations of Coputer Science, pp. 4-34, 994. [3] L. K. Grover, A fast quantu echanical algorith for database search, in Proceedings of the 8th ACM Syposiu on Theory of Coputing, pp. -9, 996. [4] L. K. Grover, Quantu Mechanical Searching, in Proceedings of the 999 Congress on Evolutionary Coputation, pp. 55-6, Jul 999. [5] L. Spector, H. Barnu, H. J. Bernstein and N. Sway, Finding a Better-than-Classical Quantu AND/OR Algorith using Genetic Prograing, in Proceedings of the 999 Congress on Evolutionary Coputation, pp , Jul 999. [6] A. Narayanan and M. Moore, Quantu-inspired genetic algoriths, in Proceedings of IEEE International Conference on Evolutionary Coputation, pp. 6-66, 996. [7] T. Hey, Quantu coputing: an introduction, Coputing & Control Engineering Journal, pp. 05-, Jun 999.

7 60 Best prots 600 Average prots Best of GQA(0) Average of GQA(0) Best of GQA() 560 Average of GQA() Best of CGA Average of CGA 530 (a) best prots (00 ites) 480 (b) average prots (00 ites) 500 Best prots 500 Average prots Best of GQA(0) 450 Average of GQA(0) Average of GQA() 40 Best of GQA() 400 Best of CGA Average of CGA (c) best prots (50 ites) 00 (d) average prots (50 ites) 900 Best prots 900 Average prots 850 Best of GQA(0) 850 Average of GQA(0) Average of GQA() Best of GQA() Best of CGA Average of CGA (e) best prots (500 ites) (f) average prots (500 ites) Figure : Coparison between CGA and GQA on the knapsack proble. The vertical axis is the prot value of knapsack, and the horizontal axis is the nuber of generations. (a), (c), (e) show the best prots, and (b), (d), (f) show the average prots. Both were averaged over 5 runs.