Quantum-Evolutionary Algorithms: A SW-HW approach

Transcription

1 Proceedngs of the 5th WSEAS Int. Conf. on COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vence, Italy, November 0-, Quantum-Evolutonary Algorthms: A SW-HW approach D. PORTO, A. MARTINEZ, S. SCIMONE, E. SCIAGURA AST Automotve STMcroelectroncs S.r.l. Contrada Blocco Torrazze, C.P. 41, 9511 Catana ITALY Abstract: - In ths paper a HW/SW platform for the mplementaton of an optmzer based on a evolutonary algorthm, called quantum-nspred evolutonary algorthm (QEA), s ntroduced. It s based on the concept and prncples of quantum computng, such as quantum bt and superposton of states, whose features are brefly descrbed. The hardware mplementaton of the QEA usng a FPGA board s therefore descrbed together wth the use of a customable software ftness n order to solve general purpose problems. HW-SW connectons are provded by a PCI nterface. Fnal am s to buld a flexble object able to optmze the choces of some sensble parameters n a dynamc system quckly, wth partcular attenton on ndustral and automotve applcatons. Key-Words: - QEA, Evolutonary Algorthms, Q-bt, Quantum computng, FPGA, Hardware Desgn. 1 Introducton In the past decade, the solutons of some complex optmzaton problems have been dealt by Genetc Algorthms, a very effectve and versatle optmzaton strategy. They work by evolvng a set of potental solutons accordng to survval rules that gve advantages to the best ndvduals. Due to ths structure, t can be avoded the fall n local maxma or mnma [1][]. The accuracy of these methodologes s often pad wth a hgh computatonal tme, especally n solvng complex problems. The recent comng of Quantum-Inspred Evolutonary Algorthm (QEA [3][4]) seems to overcome ths lmtaton, denotng the same relablty and robustness of Genetc Algorthms but faster performances. Moreover, a hardware mplementaton of QEA could provde a further acceleraton, gvng us a tool able to perform onlne optmzaton of parameters n any dynamc system, such as combuston engnes or ndustral plants. The object of ths artcle s the realzaton of a hardware to perform the several operatons composng a QEA and to mantan, however, a certan flexblty to handle problems as varous as possble. At last the algorthm wll be descrbed from a theoretcal pont of vew and t wll be shown n the mplemented detals on the board. QEA Overvew As Genetc Algorthms, QEA explots the concepts of ndvdual, populaton, evaluaton of ftness, evolutons of the populaton, hence, the dea of generaton. Snce Quantum Computng adopts the noton of Q-bt and the prncple of superposton of states, QEA, nstead of the classc bnary representaton, use a Q-bt, defned as the smallest unt of nformaton, n whch there s a coexstence of the states 1 and 0, each one wth ts probablty. The sum of the squares of these probabltes s 1. We assume a sngle ndvdual (Q-ndvdual) as a sequence of Q-bts. In ths context, a Q-ndvdual represents a lnear superposton of the states 1 and 0, n the probablstc search space. A new concept s the Q-gate, a varaton operator whch drves the evoluton towards the best soluton and towards a unque state. In fact, at the begnnng, the QEA contans a populaton of one Q-ndvdual that represents the lnear superposton of all possble states wth the same probablty. As the probablty of each Q-bt approaches ether 1 or 0 by the Q-gate, the Q-ndvdual converges to a sngle state and the dversty property dsappears gradually. Compared to classcal Genetc Algorthms, t has been seen that the QEA have better performances, for that concern processng tme. The results show that QEA performs very well even wth small populatons, wthout premature convergence as compared to the conventonal Genetc Algorthms. Fnally, we tell that QEA s not a Quantum Algorthm, but a novel Evolutonary Algorthm for a classcal computer [5], [6].

2 Proceedngs of the 5th WSEAS Int. Conf. on COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vence, Italy, November 0-, Fgure 1: Smplfed structure of QEA 1 α Lα m β1 β L β m such that α + β =1, wth =1,,, m. For example, f there s a Q-ndvdual composed by three Q-bts: then, the states of the system can be represented as: 3 Encodng and operators n QEA And now we gve some useful defntons for understandng the QEA. We already know that a number of dfferent representatons can be used to encode the solutons onto ndvduals n Evolutonary Computaton. The representatons can be classfed broadly as: bnary, numerc, and symbolc [7]. QEA uses a new representaton, called Q-bt, for the probablstc representaton of the coexstence of the nformaton 1 and 0. It s based on the physcal concept of quantum bt. We also call Q-ndvdual a strng of Q-bts. 3.1 Q-bt A Q-bt s defned as the smallest unt of nformaton n QEA. Compared to bt, dentfed unambguously by one 0 or by one 1, the Q-bt s defned wth a par of numbers (α, β), dsposed as column vector [α β] T, where α + β = 1. α gves the probablty that the Q-bt wll be found n the 0 state and β gves the probablty that the Q-bt wll be found n the 1 state. As prevously sad, a Q-bt may be n the 1 state, n the 0 state, or n a hybrd state, gven from the lnear superposton of the two states, properly weghted by factors α and β. Usng the ket-notaton, these three states can be descrbed as: = ; 1 ; =. 0 = 1 ϕ β In general, the algorthm processes Q-bt n the hybrd state. It just converges to certan values 0 or 1 at the end of the teratons. 3. Q-bt ndvdual Smlarly to genetc one, where a generc ndvdual of the populaton s represented by a strng of bt, a Q-ndvdual s represented by a strng of Q-bt, defned as: The above result means that the probabltes to represent the states 000, 001, 010, 011, 100, 101, 110, and 111 are 1/16, 3/16, 1/16, 3/16, 1/16, 3/16, 1/16 and 3/16 respectvely. By consequence, the Q-ndvdual made by three-q-bts contan the nformaton of eght states. Evolutonary Computng usng Q-bt has a better evdence of populaton dversty than other representatons, because t can represents lnear superposton of states n probablstc way. So, only one Q- ndvdual s enough to represent eght states, but n bnary representaton at least eght strngs: (000), (001), (010), (011), (100), (101), (110) and (111) are needed. 3.3 Q-gate A Q-gate s defned as a varaton operator of QEA, whose functonalty s to drve the ndvduals towards better solutons. The updated Q-bts should satsfy the normalzaton condton: α ' + β ' = 1 where α and β are the values of the updated Q-bt. An example of Q-gate s the followng rotaton gate: ' cos( θ ) sn( θ ) = ' ( ) ( ) β sn θ cos θ β Ths operator, whch provdes a rotaton of θ to the Q-bt, s the core of the algorthm and wll be dscussed n the next secton. 4 Structure of algorthm QEA works on one Q-ndvdual, t t t Q t) = q, q,..., q at generaton t. { } ( 1 m

3 Proceedngs of the 5th WSEAS Int. Conf. on COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vence, Italy, November 0-, The representaton of -th Q-bt q t, wth = 1,,, m, where m s the number of Q-bt of the Q- ndvdual, s defned as: t t q = t β From n observatons of each Q-bt of the Q- ndvdual s bult a populaton of bts, wth sze n. The structure of the algorthm s descrbed by the followng fgure. The rotaton operator U( θ ), appled at step v), s the followng: cos ( θ ) sn( θ ) ( ) ( ) ( ) U θ = sn θ cos θ where θ, wth = 1,,, m s a rotaton angle of each Q-bt toward ether 0 o 1 state dependng on ts sgn. It allows the evoluton of Q-bts, accordng to the polar plot depcted n fgure 3. Fgure 3: Polar plot of the rotaton gate Fgure : Overall structure of QEA In the followng, the algorthm QEA s shown n pseudo-code, reportng the partcular operatons performed on each step. 5 Termnaton Crtera To decde the approprate termnaton of QEA, a proper termnaton condton s necessary. Although the maxmum number of generatons s a generally used termnaton crteron n EAs, n QEA the probablty of the best soluton can be employed as a termnaton crteron thanks to the probablty representaton. The termnaton condton s desgned by usng the probablty of the best soluton b as follows: Pr ob( b) = m = 1 wth α j, p = f b = 0 j β j, f b = 1 where b s the -th bt of the best soluton b and (α, β ) s the -th Q-bt of the Q-ndvdual. The termnaton condton s defned as Prob(b) > γ 0 where 0 < γ 0 < 1. The probablty Prob(b) represents the convergence of the Q-ndvdual to the best soluton. However, snce the probablty s senstve to each Q-bt s probablty, t s not easy to set the value γ 0 : a slght dfference of γ 0 can ncrease the processng tme for a partcular problem. p j 6 Hardware mplementaton The quantum-evolutonary machne mplementng the algorthm s made by a HW/SW platform, whose scheme s showed n fgure 4.

4 Proceedngs of the 5th WSEAS Int. Conf. on COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vence, Italy, November 0-, Fgure 4: Platform HW/SW 6.1 Core organzaton The core of the Quantum Hardware s shown n the scheme of fgure 5. To descrbe the data populaton, a Dual Port SRAM memory has been used, whle, to descrbe the Q-bts, a Sngle Port SRAM memory. In the system s also present a FIFO to handle data that must be send or receve from the PC. The core contans nsde a state fnte machne to handle the operatons executed n hardware, Fgure 6 shows a scheme of ths machne. Fgure 5: CORE Scheme The PC attends to compute the populaton ftness and the Best soluton; then returns the data to the Hardware machne, arrangng the FIFO memory consequently. So the PC set to 1 the value of Start and put t n Stand by for an nterrupt. When Start s set to 1, the Core machne starts and handles the operatons. The populaton data are downloaded from the FIFO and stored n the memory (Download FIFO). Then Q-bts are updated by the evaluaton of the populaton ftness and the evaluaton of the Best (Update_Q). A new pseudo - random populaton s bult by the evaluaton of the Q-bts updated n the Q-ndvdual (Make_populaton). Fnally, data are stored n FIFO (Load FIFO) and send to PC by the nterrupt, agan. 6. Memory representaton of Q-bts The populaton data s represented from 56 elements; every element s made by four varables (x, y, z, w). 31 bts have been used to represent each varable for the data and one bt for the evaluaton (f(x, y, z, t) > b(x, y, z, w, t 1)). So, to easly represent the populaton data n hardware, t has been used a memory of 56 words of 18 bts each one. The data was stored as shown n fgure 7a. A smlar strategy has been used for representng Q- bts n hardware. Every Q-bt s defned as the ordnate couple ( ) β α, so that α + β = 1 and α, β 1. To perform effcently the Q-bts operatons, the ordnate couple ( α, β ) s bult by a couple of fxedpont values of 16 bts. So, the Q-bts are stored n a memory wth 14 words, (n fact 14 bt are necessary to represent a populaton data (x, y, z, w )) of 3 bts (31+1), as showed n fgure 7b. Fgure 6: State Machne of the CORE At every start of machne, all Q-bts n memory are ntalzed to 1 1,. From the PC t can be send the Start and Frst Start sgnals; durng the frst executon of program the Frst Start sgnal s send and the machne begn wht the constructon of frst populaton (Make_populaton) based on the observaton of the Q-bts nsde the Q-ndvdual at the value of ntalzaton. Then, data are stored n the FIFO memory and the nterrupt s pn s set to 1 to ndcate to the PC that data are ready to be used. Fgure 7a-7b: Data organzaton and Q-bts n memory 6.3 Update Q-bts The updates of Q-bts are made by a state machne, whch scheme s shown n Fgure 8. The state machne also performs the computng operatons concernng angle rotaton and rotaton of Q-bt.

5 Proceedngs of the 5th WSEAS Int. Conf. on COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vence, Italy, November 0-, Fgure. 8: Update scheme of Q-bts The state machne that handles the Q-bts update performs a hgh number of operatons. In fact for each Q-bt (ex. Q(j)) t s necessary: 1. to take the -element from the populaton;. to extract the element p(,j); 3. to update the angle rotaton theta. Only when the contrbuton of all elements of a populaton was computed ( = 0,,55 ), we can update the Q-bt and to begn the evaluaton of a new Q-bt ( ex. Q( j + 1) ). Once all Q-bts were updated, the End sgnal s set to 1, the machne goes to dle state and the Quantum Evolutonary Algorthm goes to Make_populaton phase. Fgure 9 shows the scheme of the state machne that handles the update of the Q-bts. Q-bts rotaton s done by the Rotaton module. Here t s shown the formula to perform the rotaton of Q-bts. ' ' cos( ϑ) sn( ϑ) α ( α, β ) = sn( ϑ) cos( ϑ) β In Fgure 11 s descrbed the crcut for the Rotaton module whch was employed for the rotaton of Q- bts. Fgure. 11: Rotaton Module 6.4 Make Populaton The generaton of the populaton s performed by a machne that develops pseudo-random number, startng from evaluaton of Q-bts states (see Procedure Make n secton 4). In fgure 1 s shown a scheme of Make_populaton module. Fgure 13 shows the scheme of state machne that handles the operatons. Fgure 9: Update of Q-bts FSM The compute of theta s performed by the Compute_theta module, durng the phase of Up_theta. To realze ths computaton, t was desgned a structure wth a multplexer and an accumulator, as shown n fgure 10. Fgure 1: Make_populaton scheme Fgure.10: Compute_theta Fgure 13: Make_populaton Fnte State Machne

6 Proceedngs of the 5th WSEAS Int. Conf. on COMPUTATIONAL INTELLIGENCE, MAN-MACHINE SYSTEMS AND CYBERNETICS, Vence, Italy, November 0-, When the state machne starts, by the Start sgnal, the Q-bt(0) s load from Q-bts memory. Evaluaton by the Q-bt(0) produces the element p(0,0), that represents the frst bt of the frst element of populaton P(0). Then the counter of Q-bts s ncreased, t s red Q-bt(1), and s produced p(0,1) that represents the second bt of the frst populaton element P(0). So the procedure goes on n ths way untl to the generaton of all bts n the frst populaton ndvdual. When P(0) s generated, t s wrtten n the Populaton Memory, counter P s ncreased and counter Q s reset. So t s performed the generaton of P(1), P() untl P(55). When the whole populaton s generated, the state machne sets to 1 the end sgnal and puts agan tself n watng for the start sgnal. The generaton of the populaton s pseudo-random: the Q-bt represents probablty that the new element p(,j) generated s 1 or 0. In fgure 14 s shown the Compute_pop module, that performs the generaton of element p(,j) from the evaluaton of Q-bt(j). transactons on evolutonary computaton, vol. 6, no. 6, December 00 [4] Kuk-Hyun Han and Jong-Hwan Km Quantum- Inspred Evolutonary Algorthms Wth a New Termnaton Crteron, H є Gate,and Two-Phase Scheme IEEE transactons on evolutonary computaton, vol. 8, no., Aprl 004 [5] K.-H. Han and J.-H. Km, Genetc quantum algorthm and ts applcaton to combnatoral optmzaton problem, n Proc. 000 Congress on Evolutonary Computaton. Pscataway, NJ: IEEE Press, July 000, vol., pp [6] K.-H. Han, K.-H. Park, C.-H. Lee, and J.-H. Km, Parallel quantum-nspred genetc algorthm for combnatoral optmzaton problem, n Proc. 001 Congress on Evolutonary Computaton. Pscataway, NJ: IEEE Press, May 001, vol., pp [7] R. Hnterdng, Representaton, constrant satsfacton and the knapsack problem, n Proc Congress on Evolutonary Computaton. Pscataway, NJ: IEEE Press, July 1999, vol., pp Fgure. 14: Compute_pop scheme module To generate a sequence of random numbers, t was used a module that mplements the algorthm: x n+1 = P 1 ( n,t )*x n + P (n,t ) where P 1 and P represent pseudo-random values produced by two partcular counters that freely run durng all the algorthm executon. References: [1] D. E. Goldberg: Genetc Algorthms n Search, Optmzaton and Machne Learnng. Addson Wesley, ISBN: [] Lawrence Davs Edtor Handbook of Genetc Algorthms Van Nostrand Renhold Computer Lbrary, ISBN: [3] Kuk-Hyun Han and Jong-Hwan Km, Quantum-Inspred Evolutonary Algorthm for a Class of Combnatoral Optmzaton IEEE