Quantum-Inspired Bee Colony Algorithm

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1 Opn Journal of Optmzaton, 05, 4, 5-60 Publshd Onln Sptmbr 05 n ScRs. Quantum-Insprd B Colony Algorthm Guoru L, Mu Sun, Panch L School of Computr and Informaton Tschnology, Northast Ptrolum Unvrsty, aqng, Chna Bng Branch of aqng Olfld Informaton Tchnology Company, Bng, Chna Emal: lpanch@vp.sna.com Rcvd 5 March 05; accptd August 05; publshd 5 August 05 Copyrght 05 by authors and Scntfc Rsarch Publshng Inc. Ths work s lcnsd undr th Cratv Commons Attrbuton Intrnatonal Lcns (CC BY. Abstract To nhanc th prformanc of th artfcal b colony optmzaton by ntgratng th quantum computng modl nto b colony optmzaton, w prsnt a quantum-nsprd b colony optmzaton algorthm. In our mthod, th bs ar ncodd wth th qubts dscrbd on th Bloch sphr. Th classcal b colony algorthm s usd to comput th rotaton axs and rotaton angls. Th Paul matrcs ar usd to construct th rotaton matrcs. Th volutonary sarch s achvd by rotatng th qubt about th rotaton axs to th targt qubt on th Bloch sphr. By masurng wth th Paul matrcs, th Bloch coordnats of qubt can b obtand, and th optmzaton solutons can b prsntd through th soluton spac transformaton. Th proposd mthod can smultanously adust two paramtrs of a qubt and automatcally achv th bst match btwn two adustmnt quantts, whch may acclrat th optmzaton procss. Th xprmntal rsults show that th proposd mthod s obvously supror to th classcal on for som bnchmark functons. Kywords Quantum Computng, B Colony Optmzng, Bloch Sphr Rotatng, Algorthm sgnng. Introducton Artfcal b colony algorthm s proposd as an ntllgnt optmzaton algorthm whch attmpts to smulat b colony to sarch food sourcs by scholars from Turky n 005 []. Compard wth gntc algorthm, partcl swarm optmzaton and othr ntllgnt algorthm, th outstandng advantag of ths algorthm s synchronously to prform th global and local sarch n ach of tratons, whch avods th prmatur convrgnc gratly and ncrass th probablty of obtanng th optmal soluton. At prsnt, ths algorthm has alrady bn succssfully appld to numrcal optmzaton []-[4], nural ntwork dsgn [5], dgtal fltr dsgn [6], and ntwork rconfguraton n dstrbutd systm [7], constructon of th mnmum spannng tr [8] and many How to ct ths papr: L, G.R., Sun, M. and L, P.C. (05 Quantum-Insprd B Colony Algorthm. Opn Journal of Optmzaton, 4,

2 G. R. L t al. othr flds. Howvr, th progrss has bn slow n th algorthm mprovmnt. For th dsgn of th control paramtrs, ng Haun t al. prsnt an mprovd artfcal b colony algorthm for th soluton to TSP problm [9]. Kang F t al. propos a cultural annalng artfcal b colony algorthm [0]. uan Habn t al. prsnt a nw algorthm wth applcaton of th combnaton of artfcal b colony algorthm and quantum volutonary algorthm []. As a nw computng modl, quantum computng catchs wd attnton of ntrnatonal and domstc acadmcs for ts polymorphsm, suprposton and concurrncy. At prsnt, th fuson wth gntc algorthm, mmun optmzaton, patcl swarm optmzaton and othr ntllgnt optmzaton modl s succssfully appld. In th ral quantum systm, th qubts ar dscrbd on th Bloch sphr wth two adustabl paramtrs. Howvr, n th xstng quantum ntllgnt optmzaton algorthms, ndvduals ar ncodd by qubts dscrbd by unt crcl wth a sngl adustabl paramtr. Evolutonary mchansm adopts quantum rotaton gats and quantum non-gats, and t ssntally rotats qubts about crcl cntr, whch only changs on paramtr of qubts as wll. Thrfor quantum proprts hav not bn fully rflctd. Although Rf. [] prsnts a quantum gntc algorthm basd on th Bloch coordnats of qubt, howvr, ths algorthm dos not consdr th match btwn two adustmnt quantts, whch mans t dosn t go along th shortst path whn th currnt qubt movs towards th targt qubt. As a rsult, th optmzaton prformanc s nfluncd. Ths papr proposs a nw ndvdual codng volutonary mchansm, whch s dffrnt from th codng of quantum gntc algorthm n Rf. []. In ths papr, w drctly us qubts dscrbd on th Bloch sphr to cod (not th qubts coordnats. Th volutonary sarch s achvd by rotatng th qubt about th rotaton axs on th Bloch sphr. Ths mthod can automatcally achv th bst match btwn two adustmnt quantts of b colony ndvdual qubts. For th dsgn of quantum-nsprd b colony algorthm, w laborat dsgn prncpl and mplmntaton programs of th algorthm. Takng bnchmark functons xtrmum optmzaton as xampl, t shows that th proposd mthod obvously outprforms th classcal on by comparson.. B Colony Algorthm Assumng th numbr of bs s N s, and th numbr of mployd bs and onlookr bs ar N and N u, rspctvly. Indvdual dmnson s, and ndvdual sarchng spac s S = R. Th mployd b colony s X = ( X, X,, XN and ts ntal th n-th gnraton colony s X ( 0 and X ( n, rspctvly. Th obctv functon s f : S R +. Takng mnmum optmzaton as xampl, th artfcal b colony algorthm can b dscrbd as follows: (a Randomly gnratng N s solutons, whr th -th soluton X s wrttn as: ( 0,( X = X + rand X X, ( mn max mn whr,,, X 0 by th frst N solutons; (b For th mployd b X n th currnt gnraton, th nw poston n ts nghborhood can b obtand from th followng quaton: =. Calculat th targt valus, and thn ntalz ( whr {,,, }, k { N } ( V = X + φ X X, ( k,,,, and k, and ϕ s a random numbr n th rang (0, ; (c W us grdy slcton oprator to slct th bttr ons n V and X for th nxt gnraton. Ths oprator s dnotd as T : S S, and ts probablty dstrbuton s wrttn as: s { s( X, V V} PT ( V f ( X ( f (, f < = = 0, f V (3 (d Usng th stratgy of roultt, randomly slct an mployd b, and sarch a nw poston n ts nghborhood. Ths oprator s wrttn as : N T S S, and ts probablty dstrbuton s wrttn as: s N { ( } ( s ( m PT X = X = f X f (4 m= ( Lt f bst dnot th mnmum obctv functon valu, and th corrspondng ndvdual b ( x, x,, x. W ntalz th mployd b f ts sarchng tm Bas rach to th thrshold valu Lmt and t dos not fnd a bttr poston; 5

3 G. R. L t al. (f If algorthm mt th stoppng crtra, t outputs ls go back to (b. f and th corrspondng ndvdual ( x x x bst,,,, 3. Quantum-Insprd B Colony Algorthm Ths papr studs a nw mthod of quantum-nsprd sarchng wth th fuson of b algorthm. Ths mthod s namd quantum-nsprd b colony algorthm (QIBC. 3.. scrpton of Qubts on th Bloch Sphr urng th quantum computng, a qubt s a two-lvl quantum systm whch could b dscrbd n two-dmnson complx Hlbrt spac. Basd on prncpl of suprposton, a qubt can b dfnd as: θ ϕ θ ϕ = cos 0 + sn, (5 whr 0 θ π and 0 φ π. Owng to th contnuty of θ and ϕ, a qubt can b n nfntly many dffrnt stats and dscrbd by a pont on th Bloch sphr. As shown n Fgur. 3.. Th Rotaton of Qubts about th Axs In ths papr, w crat a sarch mchansm on th Bloch sphr whch s usd to rotat th qubt about th rotaton axs to th targt qubt on th Bloch sphr. Th rotaton can smultanously chang two paramtrs of a qubt, and automatcally achv th bst match btwn two adustmnt quantts, thus w can nhanc th prformanc of optmzaton. Th ky to th abov rotaton s th dsgn of rotaton axs. Th dsgn mthod gvn by ths papr can b xprssd as th follows [3] [4]. Assumng P = px, py, p z and Q = qx, qy, q z dnot two ponts on th Bloch sphr rspctvly, th rotaton axs R axs about whch P rotats to Q along th shortst path s th vctor product of P and Q, namly, R = axs P Q. As shown n Fgur. Fgur. A qubt dscrpton on th Bloch sphr. Fgur Rotaton axs of qubt on th Bloch sphr. 53

4 G. R. L t al. Basd on th prncpl of quantum computng, th rotaton matrx maks th qubt rotat about a rotaton axs along th unt vctor n = nx, ny, n z wth radan of rotaton δ, and t s dfnd by δ δ Rn ( δ = cos I sn ( n σ, (6 whr I s th unt matrx, and σ = σ x, σ y, σ z, σ x, σ y, σ ar Paul matrcs [5]. z Thrfor, on th Bloch sphr, th rotaton matrx that rotats th ( t can b dscrbd as follows: Th rotaton opraton s gvn by whr t s traton stp QIBC Codng Mthod ( t δ ( t ( axs ( ϕ about R (, to ϕ ( t δ M = cos I sn R, σ. (7 ( t = M ( t ϕ ϕ, (8 In QIBC, th ndvdual codng adopts qubt basd on th Bloch sphr. Lt N s dnot populaton sz, dnot th numbr of varabls, and P = p( t, p( t,, pn s ( t dnot th t-th gnraton colony. urng n talzaton, th -th ndvdual can b codd as follows: whr =,,, Ns Th Procton Masurmnt of Qubts ( ( ( p (0 = ϕ 0, ϕ 0,, ϕ 0, (9 On th bass of th prncpl of quantum computng, by applyng Paul matrcs to ϕ, w can gt th Bloch coordnats of ϕ. Th qubts procton masurmnt s dnotd by th followng quatons: 0 x = ϕσ x ϕ = ϕ 0 ϕ, (0 0 y = ϕσ y ϕ = ϕ 0 ϕ, ( 0 z = ϕσ z ϕ = ϕ 0 ϕ, ( whr =,,, Ns and =,,,. In QIBC, ach qubt ar rgardd as thr paratactc gns, ach on contans thr paratactc gn chans, and ach of gn chans rprsnts an optmzaton soluton. Thrfor, ach ntty rprsnts thr optmzaton solutons at th sam tm Th Soluton Spac Transformaton In QIBC, thr optmzaton solutons gvn by ach ntty can b xprssd by Bloch coordnats. Bcaus th valu of coordnat s n (,, w must map t to solutons to th problms. Assumng th -th varabl s X Mn,Max, th soluton transformaton can b xprssd as th followng thr quatons: ( ( X = Max x Mn x + +, (3 ( ( Y = Max y Mn y + +, (4 axs b 54

5 G. R. L t al. ( ( Z = Max z Mn z + +, (5 whr =,,, Ns and =,,, Employd B Sarch Takng mnmum optmzaton as xampl, accordng to th valu of targt functon, w rank th optmzd populaton n dscndng ordr, and slct th frst N ntts to compos mployd b colony. Th rst of thm compos onlookr colony. For th -th mployd b p ( t, frst of all, randomly slct mployd bs t t k, and also randomly slct a dmnson d. Thn calculat th rotaton axs p ( and p k ( ( Raxs (, = ϕd ( t ϕ d ( t and th rotaton angl δ k ( t btwn ϕ d ( t and ϕ kd ( t ϕ kd ( t as targt, rotat ϕ d ( t about R axs (, through δ k ( t. Lt ˆ ( t aftr rotaton. By grdy slcton oprator, w slct th bttr ons btwn p ( t and ( t gnraton from th followng quatons: whr, 3.7. Onlookr B Sarch p ( t ( t, f ( ˆ ( t f ( ( t ( t, f ( ˆ ( t f ( ( t pˆ p < p = p p p ( ( = max ( ( (, ( (, ( ( ( ˆ ( ( ˆ ( ( ˆ ( ( ˆ = max,, (. Takng p dnot th mployd b p for th nxt, (6 f p t f X t f Y t f Z t, (7 ( f p t f X t f Y t f Z t. (8 It s smlar wth mployd b sarch. Frst, w calculat slctv probablty of ach mployd b wth th quaton as follows: N P( p = p ( = f p f ( p m= m. (9 To ach onlookr b, frst, w slct mployd b p accordng to th roultt mthod, and n ts nghborhood w sarch for nw poston p ˆ n th sam mthod as th mployd b sarch. If p ˆ s bttr than p, st p ˆ = p, and st th numbr of sarchs Bas = 0. If Bas s lss than thrshold Lmt, st Bas = Bas +, othrws, w ntalz p and st Bas = Algorthm Trmnaton Crtron To ths algorthm, th trmnaton crtron s th numbr of tratons. Whthr t mts th pr-st accuracy or not, th algorthm wll stop whn th lmtd numbr of tratons rachs. 4. Analyss of Exprmntal Rsults Takng functons xtrmum optmzaton as xampl, by comparng wth b colony [], Bloch quantum gntc algorthm [], lt gntc algorthm, quantum dlta potntal-basd partcl swarm optmzaton [6], w vrfy th suprorty of QIBC. All of algorthms us Matlab R009a to mplmnt on th computr (P-II.0 GHz wth.0 G mmory. 4.. Bnchmark Functons To vrfy th suprorty of QIBC, th followng tn Bnchmark functons ar usd n ths xprmnt. All of tn functons ar for th mnmum optmzaton, and X s th mnmum xtrm valu pont. ( ( ( f X x, 00 00, 0 = = = x ˆ 55

6 G. R. L t al. f ( + ( f ( = 00( x x + ( x X, 00 00, = 4 (3 f3 ( = * x ( + rand ( 0, (4 f ( (5 ( X, 00 00, 0 = x x 0. = x cos( πx = X = + +, 00 00, ( + 4( f X = + x xx, ( 5 6 = = x * x, x ( k (6 f6 ( X = cos( yk +, yk 00( xk x ( x ( = 0 k= = y 4000 (7 f ( = g( x, x + + g( x, x + g( x, x 7 = +, ( 0 f X = ; = +, 00 00, x =, X, g( x y ( x y ( x y 00 x 00, 0 (8 ( = = = f8 X = x + x + x, 00 x 00, (9 f ( X = x cos +, 500 x = = x ( (0 f ( X = 0+ y 0 cos( πy * ( f X = 0. 0 = 4.. Paramtrs Sttng, x ( , = + sn ,, 0 * x =, ( f X = 0 ; x, x < y =, 00 x 00, x = 0, round ( x, x For convnnt comparson, basd on th complxty of bnchmark functons, w st up prcson thrsholds ε for ach functon. Only whn optmzaton ffct s lss than th thrsholds, th algorthm s call convrgs. In ths xprmnt, th prcson thrsholds ar st as follows. for f, f 6, ε =.0 ; for f ε = 0 ; for f 3, ε = 0 ; for f 4, f 9, f 0, ε = 0 ; for f 5, ε = 0 ; for f 7, ε = 0 ; for f, 4 8 ε = 0. If th algorthm convrgs, th currnt optmzaton stps ar calld th numbr of traton. If not, th numbr of traton s qual to th pr-st lmtd numbr of tratons. Th dmnsons of tn functons ar st to = 30, and populaton szs of fv algorthms ar qual to 40. For QIBC and BC, w hav N = 0, N u = 0, Lmt = 50. For BQGA, on th bass of Rf. [], th ntal valu of rotaton angl s 0.05π, and th mutaton probablty s 0-3. For QPSO, basd on Rf. [6], th control paramtr s λ =.. For EGA, th crossovr probablty s 0.8, th mutaton probablty s 0.0. Th lmtd numbr of tratons of fv algorthms s G = Comparson and Analyss of Smulaton To nhanc th obctvty of comparsons, ach algorthm ndpndntly runs 50 tms, and w hav comparsons of statstcal rsults. Th avrag tm for ach of traton ar shown n Tabl. To mak comparsons b suffcnt, bsds showng th avrag rsults, th bst and th worst ons ar also ar gvn. Comparson of th numbr of tratons, avrag tratons, and optmzaton rsults ar shown n Tabl. From Tabl, for th runnng tm, w rank fv algorthms n dscndng ordr as QIBC, BQGA, BC, EGA, QPSO. Th rason s that, n QIBC and BQGA, w us codng mchansm of qubt wth thr chans. urng ach traton, ach ntty rspctvly do updatng, soluton to spac transformaton, calculatng valu of bnchmark functons tc. and ths opratons tak a long tm. QIBC s longr than BQGA bcaus QIBC nds th procton masurmnt, dsgn of rotaton axs and rotaton matrcs, and ts calculatng quantty s 56

7 G. R. L t al. Tabl. Comparson of avrag tm for ach traton (unt: scond. No. QIBC BC BQGA QPSO EGA f f f f f f f f f f Tabl. Comparson of optmzaton rsults n fv algorthms. No. f Algorthm Convrgnc Numbr Avrag Itraton Avrag Rsults Th Worst Rsult Th Bst Rsults QIBC BC 0 0, BQGA 0 0, QPSO EGA 0 0, QIBC BC f BQGA 0 0, QPSO EGA 0 0, , QIBC BC f3 BQGA 0 0, QPSO EGA 0 0, QIBC BC f4 BQGA 0 0, QPSO EGA 0 0, QIBC BC f5 BQGA 0 0, , QPSO EGA 0 0,

8 G. R. L t al. Contnud QIBC BC f6 BQGA 0 0, QPSO 0 0, EGA 0 0,000,7 355, QIBC BC f7 BQGA 0 0, QPSO EGA 0 0, QIBC BC 0 0,000 50, 69,884 33,93 f8 BQGA 0 0, , QPSO EGA 0 0, , QIBC BC f9 BQGA QPSO EGA 0 0, QIBC BC f0 BQGA 0 0, QPSO EGA 0 0, largr. In Tabl, for th ablty of optmzaton, obvously, QIBC s bttr than QIBC, BQGA s bttr than EGA. QPSO s wors than QIBC, but t s bttr than BQGA, and t s smlar to BC. For th optmzaton prformanc of fv algorthms, w rank thm n dscndng ordr as QIBC, BC, QPSO, BQGA, EGA. For th abov rsults, w prsnt th followng analyss. Frstly, w ntroduc th codng mchansm of qubt wth thr chans, and t ffctvly nhancs th rgodcty of algorthm to soluton spac. Basd on th gomtrc proprts of th Bloch sphr, ths mchansm can nlarg th numbr of global optmal solutons and th probablty of attanng global optmal solutons. That s th rason why ths two algorthms ar bttr than thr classcal ons rspctvly. Scondly, QIBC has a bttr prformanc of optmzaton than BQGA, bcaus thy adust qubt n th dffrnt ways. In BQGA, qubts ar updatd by drctly adustng θ and ϕ wth th sam adustmnt amount (namly θ = ϕ. Obvously, n ths mthod, t s hard to clos to targt qubts along th shortst path, bcaus thr must b som matchng rlaton btwn θ and ϕ whn t s along th shortst path. But ths matchng rlaton s hard to b xprssd clarly by analytc quatons. In QIBC, w adust qubt n ndrct mthod. Namly, w rotat qubt to targt qubt about th rotaton axs on th Bloch sphr, and obvously, ths path s 58

9 G. R. L t al. Tabl 3. Comparson of optmzaton rsults n QIBC and BC. Functon f9 f0 Algorthm Convrgnc Numbr Avrag Numbr Avrag Rsults Th Worst Rsult Th Bst Rsult QIBC BC 0 0, QIBC BC th shortst. Although t dos t adust two paramtrs of qubt drctly, t achvs th bst match btwn θ and ϕ automatcally and accuratly. Hnc, ths mthod has bttr optmzng ffcncy than classcal ons. It s also worth pontng out that, although th QIBC has bttr optmzng prformanc, th complxty of ths algorthm s obvously hghr than classcal ons. Comparng wth classcal ons, th QIBC nds som xtra opratons, such as calculatng rotaton axs, rotaton angl, rotaton matrcs and th soluton to spac transformaton. Consdrng run tm and optmzng prformanc, QIBC arns bttr optmzng ffcncy by sacrfcng runnng tm, whch s th sam as No Fr Lunch. For many off-ln optmzng tasks whch may gnor runnng tm, th QIBC has wd applcaton prospcts. W nd nvstgat and study th nflunc aftr dmnsons and paramtrs chang. Takng f 9 and f 0 4 as xampl, whl = 00, populaton sz s N = Nu = 40, and Lmt = 00, G = 0. Prcson thrshold 0 of f 9 and f 0 s 0. Th QIBC and th BC run 0 tms rspctvly. Optmzaton rsults s shown n Tabl 3. As th rsults ar shown, for th hgh-dmnsonal optmzaton problms, th proposd algorthm also shows ts bttr prformanc than classcal ons. 5. Concluson W prsnt a quantum-nsprd b colony optmzaton algorthm. In proposd mthod, th bs ar ncodd wth th qubts dscrbd on th Bloch sphr. Th populaton volutonary s achvd by rotatng th qubt about th rotaton axs on th Bloch sphr. Th xprmntal rsults show that, th mthod of qubts codng on th Bloch sphr and th mthod of b updatng whch can achv th bst match btwn two adustmnt quantts of qubt paramtrs by rotatng about th rotaton axs, can truly nhancs th prformanc of th ntllgnt optmzaton algorthm. Acknowldgmnts Ths work was supportd by th Natural Scnc Foundaton of Hlongang Provnc, Chna (Grant No. F050, th Youth Foundaton of Northast Ptrolum Unvrsty (Grant No. 03NQ9 and th Scnc Tchnology Rsarch Proct of Hlongang Educatonal Commtt of Chna (Grant No Rfrncs [] Karaboga,. (005 An Ida Basd on Hony B Swarm for Numrcal Optmzaton. Tchncal Rport-TR06, Engnrng Faculty, Computr Engnrng partmnt, Ercys Unvrsty, Kaysr. [] Bahry, A. and rvs, K. (0 A Modfd Artfcal B Colony Algorthm for Ral-Paramtr Optmzaton. Informaton Scncs, 9, [3] Xang, W.L. and An, M.Q. (03 An Effcnt and Robust Artfcal B Colony Algorthm for Numrcal Optmzaton. Computrs & Opratons Rsarch, 40, [4] L, G.Q., Nu, P.F. and Xao, X.J. (0 vlopmnt and Invstgaton of Effcnt Artfcal B Colony Algorthm for Numrcal Functon Optmzaton. Appld Soft Computng,, [5] Karaboga,., Akay, B. and Ozturk, C. (007 Artfcal B Colony (ABC Optmzaton Algorthm for Tranng Fd-Forward Nural Ntworks. Modlng csons for Artfcal Intllgnc, 467,

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