A Hierarchical Approach to Computer-Aided Design of Quantum Circuits

Transcription

1 A Hierrhil Approh to Computer-Aided Design of Quntum Ciruits Mrek Perkowski,+* Mrtin Luk,* Mikhil Pivtoriko,* Pwel Kerntopf, & Mihele Folgheriter *, Dongsoo Lee, + Hyungok Kim,+ Woong Hwngo, Jung-wook Kim+ nd Yong Woo Choi.+ *Deprtment of Eletril nd Computer Engineering, Portlnd Stte University, Portlnd, OR, USA, +Deprtment of Eletril Engineering, Kore Advned Institute of Siene nd Tehnology, 373-1, Kusong-dong, Yusong-gu, Tejeon, , Kore, & Institute of Computer Siene, Wrsw University of Tehnology, Nowowiejsk 15/19, Wrsw, Polnd. 1. Introdution Astrt: A new pproh to synthesis of permuttion lss of quntum logi iruits hs een proposed in this pper. This pproh produes etter results thn the previous pprohes sed on lssil reversile logi nd n e esier tuned to ny prtiulr quntum tehnology suh s nuler mgneti resonne (NMR). First we synthesize lirry of permuttion (pseudoinry) gtes using Computer-Aided-Design pproh tht links evolutionry nd omintoris pprohes with humn experiene nd retivity. Next the iruit is designed using these gtes nd stndrd 1*1 nd 2*2 quntum gtes nd finlly the optimizing tutologil trnsforms re pplied to the iruit, produing sequene of quntum opertions prtilly relizle. These hierrhil stges n e ompred to stndrd gte lirry design, generi logi synthesis nd tehnology mpping stges of lssil CAD systems, respetively. We use n informed geneti lgorithm to evolve ritrry quntum iruit speified y (trget) unitry mtrix, speifi enoding tht redues the time of lulting the resultnt unitry mtries of hromosomes, nd n evolutionry lgorithm speilized to permuttion iruits speified y truth tles. We outline intertive CAD pproh in whih the designer is prt of feedk loop in evolutionry progrm nd the serh is not for iruits of known speifitions, ut for ny gtes with high proessing power nd smll ost for given onstrints. In ontrst to previous pprohes, our methodology llows synthesis of oth: smll quntum iruits of ritrry type (gtes), nd permuttion lss iruits tht re well relizle in prtiulr tehnology. While quntum mehnis nd quntum omputing re estlished reserh res, utomted quntum iruit synthesis is still only t the eginning of its explortion

2 [2,4,6,7,8]. In quntum omputtion we use quntum its (quits) insted of lssil inry its to represent informtion. This gives the dvntge of eing le to perform mssively prllel omputtions in one time step. The design of quntum iruits of prtil size is still tehnologilly impossile (the mximum numer of quits in yer 2002 is 7), ut the progress is fst nd there re no rguments sed on physis ginst the possiility of uilding powerful quntum omputers in the future. Therefore quntum omputing re of reserh is reently flourishing. Finding n effetive nd effiient method of designing quntum iruits n hve three pplition res: (1) Optimizing quntum iruits for NMR [25,26,27,28,29], ion trp, quntum dot, vity quntum eletrodynmis or Si-sed nuler spin quntum omputer tehnologies tht exist lredy in prtie. Eh of these tehnologies hs different minimiztion requirements nd ll of them require minimizing oth the width nd the length of srthpd register (numer of quits proessed). Width is solutely ritil nd length is lso importnt euse of deoherene. Ciruit redution is very importnt for present tehnology, so ext or su-miniml methods should e developed for smll iruits, less thn 7 vriles. This is very smll numer from the stndrd CAD lgorithms point of view, ut the synthesis prolem for quntum logi is more diffiult. (2) Modeling quntum omputers in FPGA-sed reonfigurle hrdwre for speeding-up omputtions tht re very ineffiient on stndrd omputers [9]. Sine for urrent FPGA tehnologies only smll quntum iruits n e emulted, the requirements re similr to point (1) (3) Designing new optimized gtes nd iruits for theoretil investigtions nd for use in future quntum omputers. If we elieve tht suh omputers will exist, effiient CAD methods for lrge numer of vriles, tens or hundreds, should e developed, ut these lgorithms will e non-optiml. This is theoretil reserh for non-existing tehnology in yer Quntum Computing The mjor differene etween quntum logi nd inry logi is the onept of the informtion itself. While the lssil (inry or multi-vlued) representtions of informtion re preise nd deterministi, in Quntum Computing (QC) the onept of it is repled y the quit. Unlike lssil its tht re relized s eletril voltges or urrents present on wire, quntum logi opertions mnipulte quits [7]. Quits re mirosopi entities suh s photon or tomi spin. Boolen quntities of 0 nd 1 re represented y pir of distinguishle different sttes of quit. These sttes n e photon s horizontl or vertil polriztion denoted y > or >, or n elementry prtile s spin denoted y > or > for spin up nd spin down, respetively. After enoding these distinguishle quntities into Boolen onstnts, ommon nottion for quit sttes is 0> nd 1>. Quits exist in liner superposition of sttes, nd re hrterized y wvefuntion ψ. As n exmple, it is possile to hve light polriztions other thn purely horizontl or vertil, suh s slnt 45 orresponding to the liner superposition of ψ=½[ 2 0>+ 2 1>]. In generl, the nottion for this superposition is α 0>+β 1>. These intermedite sttes nnot e distinguished, rther mesurement will yield tht the quit is in one of the sis sttes, 0> or 1>. The proility tht

3 mesurement of quit yields stte 0> is α 2, nd the proility is β 2 for stte 1>. The solute vlues re required sine, in generl, α nd β re omplex quntities. Pirs of quits re ple of representing four distint Boolen sttes, 00>, 01>, 10> nd 11>, s well s ll possile superpositions of the sttes. This property is known s entnglement, nd my e mthemtilly desried using the Kroneker produt (tensor produt) opertion [7]. As n exmple, onsider two quits with ψ 1 =α 1 0>+β 1 1> nd ψ 2 =α 2 0>+β 2 1>. When the two quits re onsidered to represent stte, tht stte ψ 12 is the superposition of ll possile omintions of the originl quits, where ψ 12 = ψ 1 ψ 2 = α 1 α 2 00> + α 1 β 2 01> + α 2 β 1 10> + β 1 β 2 11>. (1) Superposition property llows quit sttes to grow muh fster in dimension thn lssil its. In lssil system, n its represents 2 n distint sttes, wheres n quits orresponds to superposition of 2 n sttes. Oserve lso tht in the ove formul some oeffiient n nel, so there exist onstrint ounding the possile sttes in whih the system n exist. These ll ontriute to diffiulties in understnding the onepts of quntum omputing nd reting effiient nlysis, simultion, verifition nd synthesis lgorithms for QC. Generlly, however, we elieve tht muh n e lerned from the history of Eletroni Computer Aided Design nd the lessons lerned should e used to design effiient CAD tools for quntum omputing. In terms of logi opertions, nything tht hnges vetor of quit sttes n e onsidered s n opertor. These phenomen n e modeled using the nlogy of quntum iruit. In quntum iruit wires do not rry Boolen onstnts, ut orrespond to pirs of omplex vlues, α nd β. Quntum logi gtes of this iruit mp the omplex vlues on their inputs to omplex vlues on their outputs. Opertion of quntum gtes is desried y mtrix opertions. Any quntum iruit is omposition of prllel nd seril onnetions of loks, from smll to lrge. Seril onnetion of loks orresponds to multiplition of their (unitry) mtries. Prllel onnetion orresponds to Kroneker multiplition of their mtries. So, theoretilly, the nlysis, simultion nd verifition re esy nd n e sed on mtrix methods. Prtilly they re tough euse of the prolem dimension - exponentil growth of mtries. Synthesis prolem n e formulted s deomposing hierrhilly given unitry mtrix to seril nd prllel onnetions of smller mtries, until si diretly relizle quntum primitives re rehed. This prolem is very diffiult in suh si formultion nd therefore severl speil methods hve een nd re eing developed, espeilly in the lst 5 yers. Proilisti lultions sed on this representtion re used in only very smll quntum omputers so fr (most with 3 its), ut it ws verified tht informtion n e represented s superposition of sttes of single quits, nd tht in one time step opertions n e performed on severl quits. Beside this useful effet of quntum omputing, vrious other effets resulting from quit enoding emerge, suh s quit entnglement. Moreover it ws shown [7] tht ny quntum omputing hs to e reversile, whih ffets ll synthesis methods. Conluding, uilding quntum omputers eomes more nd more tehnil rther thn only sientifi issue, nd the methods developed to design them, suh s forml representtion, modeling nd synthesis will hve pplitions not only to quntum omputing ut lso to DNA nd other nno-tehnologies euse of their reversile nture.

4 In this pper we fous only on the synthesis of ritrry quntum iruits (nd quntum gtes in prtiulr) of smll size, less thn five vriles. We onentrte on designing the iruits from lss of permuttion gtes whih hve unitry mtries eing permuttion mtries. Suh iruits orrespond to lssil Boolen funtions. However, our gte design methods re for generl quntum iruits, with ritrry unitry mtries. The presented methods n e speilized to some lsses suh s f- ontrolled-phse-shift gte iruits [25], nd generlized to multiple-vlued quntum logi [30,41] nd mixed multiple-vlued logi. The pper is orgnized s follows. First, we disuss some importnt issues relted to quntum logi iruits. By disussing them we wnt to prove tht it is not true tht urrent inry reversile logi synthesis methods n e diretly pplied to permuttion quntum iruits. (This is ommon elief, prtilly true, ut only first step). By presenting this disussion, we would like to enourge reserhers with logi synthesis kground to rete new improved methods tht will e more prtilly useful to optimizing sequenes for urrent existing tehnologies suh s NMR. Our theses in the first prt re the following: 1. Toffoli (CCNOT) nd Fredkin re not lwys the est gtes for quntum omputing. (Toffoli gte is desried y equtions P=,Q=, R=, Fredkin gte y equtions P=, Q= +, R=+ ). Multi-input Toffoli gte look simple in digrm, ut tke lot of gtes when redrwn to 3*3 gtes with uxiliry onstnts - they re not primitives. The synthesis should e performed t one hnd on lower level of quntum primitives suh s CNOT (Feynmn), ontrolled squre-root-of-not nd Hdmrd, nd on the other hnd on the level of more powerful reversile gtes, suh s those introdued in [17,35], to simplify the serh y inresing gte grnulrity. 2. The trnsformtion from optiml reversile iruit (on ny level of gtes) to the miniml quntum sequene for NMR progrmming is fr from eing trivil, nd so fr no omintoril optimiztion lgorithms hve een reted for them. The prolems of gte ordering, input vrile ordering, lol trnsformtions, removl of swp gtes (plnriztion) nd others, re quite similr to tehnology mpping nd physil design res in lssil CAD, ut so fr they re solved only d ho y physiists [43]. 3. All gte ost ssumptions tht n e found in generl quntum ppers (nd not in speilized ppers out NMR tehnology) re fr too pproximte nd n led to highly non-optiml sequenes. Seond, fter presenting these diffiulties in more detil, we outline our CAD tools, in whih evolutionry nd humn-oriented intertive methods re omined. We propose generlized pproh to the prolem of quntum omputing (QC) CAD y using simple enoding nd generi geneti lgorithm (GA) without ny prolemspeifi opertors. Our results show tht, in ontrst to pulished work [4,6], ny kind of geneti opertors n e used. We were le to synthesize ompletely utomtilly more omplex iruits thn those y previous progrms, for instne the Toffoli nd Mrgolus gtes. 3. New Gtes nd their Cost Funtions for the Optimiztion Algorithms

5 Figure 1 To + An importnt prolem, not disussed so fr y other uthors, is the seletion of the ost funtions to evlute the quntum iruit designs. Although the detiled osts depend on ny prtiulr reliztion tehnology of quntum logi, so tht the ost reltions etween for instne Fredkin nd Toffoli gtes n differ in NMR nd ion trp reliztions, the ssumptions used in severl previous ppers; tht eh gte osts the sme, or tht the ost of gte is proportionl to the numer of inputs/outputs, re oth fr too pproximte. In this pper we will illustrte more preise ost funtions for gtes tht re used in our optimiztion lgorithms (even more urte osts will e presented in the forthoming pper, here we just wnt to signlize the importnt prolem). We will follow in the footsteps of previous ppers in quntum omputing [12,15,25-29] nd we will relize ll gtes from 1* 1 nd 2*2 gtes. Moreover, ording to [15] we will ssume tht the ost of every 2*2 gte is the sme. We will ssume lso tht 1*1 gte osts nothing, sine it n e lwys inluded to ritrry 2*2 gte tht preedes or follows it. Thus, in first pproximtion, every permuttion quntum gte will e uild from 1*1 nd 2*2 quntum primitives nd its ost lulted s totl sum of 2*2 gtes used. Using the well-known reliztion of Toffoli gte with truly quntum 2*2 primitives, shown in Figure 1 [15], the ost of Toffoli gte is 5 2*2 gtes, or simply, 5. In Figure 1, is squre-root-of-not gte (unitry mtrix ) nd + is its hermitin. Thus retes unitry mtrix of NOT gte nd + = I (n identity mtrix, desriing just quntum wire). The reder n nlyze orretness of this onstrution y nlyzing ll possile vlues of inputs signls. (the generliztion of this gte to n- inputs without onstnt wires is shown in [12]). Now we will relize the Fredkin gte from the Toffoli gte. Using GA [16] or synthesis methods from this pper, we n synthesize the Fredkin gte using two Feynmn nd one Toffoli gte s in Figure 2.

6 Figure 2 =.. To.. ( ) = = Sustituting the Toffoli design from Figure 1 to Figure 2 we otin the iruit from Figure 3. After using the ovious EXOR-sed trnsformtion shown in the right (hnge of order of Feynmn gtes), the finl iruit from Figure 3 is otined. Oserve tht sde of two 2*2 gtes is nother 2*2 gte, so following [15] we otin iruit from Figure 3 with the ost of 5. (Eh sequene in dotted lines hs ost of 1). Thus, the ost of Toffoli gte is extly the sme s the ost of Fredkin gte, nd not hlf of it, s some uthors ssumed nd s my e suggested y lssil inry shemt of suh gtes, where the Toffoli gte inludes single Dvio gte, while the Fredkin gte inludes two multiplexers. Enourged with this oservtion, let us lulte osts of other known gtes. It ws shown in [31] tht the ost of Miller s Gte is 7 nd not 9, s might e expeted from its inry shemtis using Feynmn nd Toffoli gtes. Interestingly, nother reliztion of Miller s gte hs n even smller ost of 6. Oserve tht sine swp gte n e relized y sde of three Feynmn gtes, ording to the ost evlution method from [15] its ost is lso 1 nd not 3 s ssumed previously. Similrly the osts of 3*3 gtes y Kerntopf [24,34,37], Mrgolus [21], De os [24], Khn [20], nd Mslow [21,22], osts of ll 4*4 Perkowski s gtes [17], nd other gtes from [24] n e lulted. Next oserve tht new permuttion quntum gte with equtions: P = Q = R = n e relized with ost 4. It is just like Toffoli gte from Figure 1 ut without the lst Feynmn gte from right. This is the hepest quntum reliztion known to us of omplete (universl) permuttion gte nd it is thus worthy further

7 investigtions. We found tht the eqution of this gte ws known to Peres [21], ut it hs een not used in reversile or quntum omputing. Oserve tht lgorithms from [33,40], when given the eqution of Peres gte, would return the solution omposed fro Toffoli nd Feynmn gtes, whih would led to lerly non-miniml quntum sequene. However, if the postproessing stge of mrogenertion nd next equivlene simplifying trnsformtions were pplied to results from [33,40], the minimum solution would e found. It is now hllenge to other reserhers to find heper universl 3*3 gte, tht would use only 2*2 nd 1*1 gtes (it is known tht severl suh universl sets of gtes exist. [12]). Figure 3 ) + ) + Figure 4 Cost = four 2*2 quntum gtes + + = (+) C=0 ==> (+) C=0 ==> (*)

8 Oserve in Figure 4, tht the Peres Gte with Feynmn gtes loted in ll possile pirs of terminls t its periphery retes powerful nd inexpensive new gtes. For instne y loting Feynmn gte with EXOR up on output wires nd, genertes gte with equtions: =, = (+), =. Agin, the Feynmn gte n e omined with the + gte so the ost of the omined new gte is gin only four. Another possiility of onneting Feynmn gte to the outputs, of the Peres gte is shown in Figure 5. There exist lso two possiilities of onneting Feynmn gte to inputs,, whih similrly led to two new gtes, nd four possiilities of onneting Feynmn gtes to oth inputs, nd outputs,, whih leds to four new gtes. Severl other new gtes n e reted for nother input/output terminl pirs in Figure 4. Figure 5 Cost = four 2*2 quntum gtes ) + + = (+) C=0 ==> (+) C=1 ==> (+) = 4. Frme-sed serh genertor nd geneti lgorithms Let us oserve tht there exist very mny omintoril possiilities of onneting Feynmn (nd other) gtes to inputs nd outputs of si gtes, s ws shown in setion 3. We me thus to onlusion tht suh proedure should e utomted. Our frme-sed serh genertor llows exhustive nd prtil serh, nd it is intertively lled y the user, who delres struture, gtes, onstrints nd serh prmeters like depth, serh type (depth first, redth first, OR vs AND/OR, et), numer of node hildren, et. [42]. The user selets some seed gte nd defines symolilly ll possile strutures tht n e uild y onneting dditionl gtes to it. The types of these gtes, the terminls of their utment nd the simplifying (omining) trnsformtions to e pplied, s well s osts of gtes, re the prmeters of the progrm. These gtes re like slots in frme struture used in AI progrmming. The sequene of slots retes GA s hromosome. This new method, whih we ll informed GA, omines evolutionry progrmming nd frmes. The detils out evolutionry lgorithms for oth truly quntum iruits nd permuttion

9 iruits n e found in [16] nd [17], respetively. All utomtilly reted new gtes, or gtes stisfying some riterion (like smll ost or (prtil) stisftion of unitry mtrix) re displyed on sreen to the user, together with their ost funtions. This wy, lirry of powerful omplex permuttion gtes is reted tht on one hnd re gered towrds strutured design methods (suh s Kerntopf gte used in regulr strutures lled Nets [24,37,39], nd on the other hnd the gtes tht hve good reliztions (short nd ompletely relizle sequenes) in ny prtiulr quntum tehnology, suh s espeilly NMR [25-29]. Next the mrogenertions from higher order permuttion quntum gtes (3*3, 4*4, nd higher) to quntum gte primitives re pplied. They re followed y tutology nd equivlene trnsformtions not on the level of permuttion gtes (s in [33,40]) ut on the level of truly quntum 1*1 nd 2*2 primitives. This llows to optimize the quntum iruits further. For instne, n n-input Toffoli gte n e uild without onstnt inputs using only 2*2 primitives s result of mrogenertion [12]. These trnsformtions re in priniple similr to those presented in detil in [25-29,33,40] nd will e not disussed here. Mny of them re sed on ovious rules of EXOR lger, suh s some illustrted ove. Enoding for the frme genertor is the sme s for the geneti lgorithm. Also, the sme mtrix-opertion sed verifition sheme from the fitness evlution suroutine of GA is used. An exmple of our enoding is shown in Figure 6. 1 W H 3 W H H S W S S S W W W 2 W Figure 6: Trnsformtion of QC from the enoded hromosome (on the left) to finl quntum iruit nottion representtion of this QC (on the right). Here S is Swp gte, H is Hdmrd gte, W is wire. In the middle there is one CCNOT (Toffoli) gte. On the left side of Figure 6 it is shown how the iruit on the right of the sme figure is enoded. As n e seen, there is no free spe in the proposed enoding. Eh ple in the iruit is presented s symol of unitry mtrix of ertin elementry quntum gte. A wire hs unitry identity mtrix representtion. While evluting the fitness funtion, Kroneker produts (tensor produts) re exeuted on mtries of prllel gtes (loks, iruits), nd stndrd mtrix multiplitions re performed on seril onnetions of gtes. Eh QC is prsed in prllel loks, evluted seprtely nd finlly multiplied together to give the finl unitry mtrix representtion of the QC. This finl mtrix is next ompred with the trget mtrix to evlute their

10 distne s prt of fitness funtion lultion. The exmple shown in Figure 6 illustrtes ommon inonveniene of enoding quntum iruits for geneti lgorithms. A quntum gte CCNOT n e pled over three different ritrry wires in quntum iruit. However with the enoding used, there is no informtion inditing wht gtes re onneted to wht wires, eside the order of the gtes. To solve this prolem we insert two Swp gtes (one efore nd one fter) the CCNOT. This implies tht outside of the Swp gtes the CCNOT seems like eing on wires 2,4 nd 5, ut the rel CCNOT gte uses wires 3,4 nd 5. In order of eing le to enode QC without ny dditionl prmeters, the iruit is split into prllel loks where eh lok n e evolved or intertively mnipulted seprtely. Figure 7: Exmples of Kroneker produt, nd of Mtrix produt * on smple of iruit. 5. Experimentl results The results re quite enourging. In every se the GA found the requested gte, however in no se the utomtilly reted hromosome ws etter thn the iruit for whih the orresponding trget unitry mtrix ws reted. Summry of results is shown in Tle 1. Numer of inputs per q-gte Numer of genertions pm pc Rel time (verge 20 runs) pm<0.2 Numer of genertions Rel time (verge 20 runs) Popultion size 1 - input < < 30 seonds <100 < 1 minute inputs < < 30 seonds <100 < 1 minute inputs <1 minutes <200 <3 minutes 60

11 Tle 1: Results of experiments. Due to the similrity of results we grouped the results y the numer of inputs/outputs of the requested q-gte. PM nd PC re proility of muttion nd rossover. All iruit evolved were ext opies or t lest hd sme numer of wires, of the serhed iruit. For smll gtes the onvergene is logilly fster, euse of the restrited reomintion etween different gtes. A gte with more inputs thn the numer of wires in the iruit ws not tested. The 3 input gtes test shows the sme result, however with exponentil time. The results re mesures of verge vlues over 20 runs. Depending on the iruit we were looking for, the times re, s predited, inresing very fst with the inrese of the numer of QC inputs. Two onfigurtions were tested. First, high proilities of muttion ( ) nd rossover ( ) were pplied. The results re surprising euse of so high muttion proility. The size of the GA popultion ws set in rnge [50,100]. The lol very high proility of muttion llows fst dngerous serh. However the runs were stopped s soon s good solution ws found nd the est individul exmined. A lrge rndom serh with muttion used on reomintion prolem seems to hve positive effet in restrited serh. The solution ws found lso when the muttion ws of smll order, however the time of serh rised s well. Next step ws to test omposite iruits proposed y [4,6]. We seleted three of them shown in Figure 8. The first two re oth iruits to produe EPR s in [4], the lst is the send iruit originlly proposed y [10] nd evolved in [6]. H H H H Figure 8: Three types of enhmrk iruits from literture serhed with the GA. The results re shown elow in Tle 2. We were le to find ll serhed iruits, however in this prt of experimenttion the strting set of gtes ws open. Our GA found for ll enhmrks t lest similr, if not etter, results ompred to the pulished results of the studied ses. Even if the numer of genertions grows exponentilly, the rel time still remins resonle.

12 Numer of inputs per q- gte Numer of genertions pm pc Rel time (verge 20 runs) pm<0.2 Numer of genertions Rel time (verge 20 runs) Popultion size 3 - input < < 1 minutes <300 < 2 minute inputs < < 2 minutes <900 < 3 minute 50 Tle 2: Results of enhmrk tests for ssemled iruits. The 3- nd 4- input iruit serh ws mde under similr onditions s the first prt of experiments. Results from oth tles shows tht GA n e quite suessfully used to synthesize iruits. The time n e redued y pproprite hrdwre nd onsequently used for still lrger designs. Using this lgorithm, we were not le to find less expensive quntum reliztions of ny 3*3 permuttion gtes thn Smolinlike reliztions, well-known solutions nd our hnd designs, ut we found the sme or new gtes of the sme ost (the optimized version of the send iruit found y Willims, Toffoli nd Mrgolus gtes). For instne, only 99 individuls were reted in the geneti lgorithm pool to find the optiml solution to Mrgolus gte. Mny new gtes, some interesting nd of smll osts, hve een found s y-produt of serhing for known gtes. These results will e nlyzed elsewhere. 6. Conlusion We hve shown tht the hierrhil pproh n e used for utomted QC development using stndrd PC omputers. Our progrm found one new iruit tht ws erlier me ross y Willims [7], three iruits loted y Ruinstein [4] nd severl new iruits. In ll ses tht we studied the progrm ws fster thn the results previously pulished. In ontrst to previous works tht onentrted on some prtiulr types of iruits suh s teleporttion [7] nd entnglement [4] our pproh is fully generl. It is lso hierrhil nd llows to inlude humns in the design loop. We will further experiment with the lgorithm trying to find vrious reliztions for gtes nd iruits from the literture ited elow. The lgorithm from [17] is pplied to permuttion iruits. Suh iruits re used for instne in the fmous Grover s Quntum Serh Algorithm [11]. Reversile gtes relizing Boolen opertions n e relized not in quntum ut in severl other reversile tehnologies suh s DNA, single-eletron trnsistor, mehnil nnoswithes, quntum dots or CMOS. Although ll enhmrks proved the onvergene of our GA nd results were etter thn previous ones, the gol of our pproh is not only to enhmrk GA, ut minly to explore vrious evolutionry nd other pprohes [18,21,22,25-29,35,36] to reversile nd quntum iruit synthesis. Next step is to pply different Evolutionry lgorithms suh s Bldwinin or Lmrkin GA, geneti engineering or Evolutionry strtegies. We rete enhmrk lirry to e used y other interested QC reserhers. We lso ontinue to work on improvements to the progrms nd new experimentl results will e shown during the workshop presenttion.

13 7. Referenes 1. D. E. Golderg, Geneti Algorithms in Serh, Optimiztion, nd Mhine Lerning Addison Wesley, Y. Z. Ge, L. T. Wtson, nd E. G. Collins. Geneti lgorithms for optimiztion on quntum omputer. In Unonventionl Models of Computtion, pp K-H Hn, K-H Prk, C-H Lee, nd J-H Kim, Prllel quntum-inspired geneti lgorithm for omintoril optimiztion prolems, In Proeedings of the 2001 Congress on Evolutionry Computtion, volume 2, pp , B.I.P. Ruinstein, Evolving quntum iruits using geneti progrmming, Proeedings of the 2001 Congress on Evolutionry Computtion (CEC2001), pp (2001) 5. L. Spetor, H. Brnum, H. J. Bernstein, nd N. Swmy, Finding etter-thnlssil quntum AND/OR lgorithm using geneti progrmming, In Proeedings of the 1999 Congress on Evolutionry Computtion, volume 3, pp , Wshington D.C., July IEEE, Pistwy, NJ. 6. C.W. Willims, Gry G. Alexnder, "Automted Design of Quntum Ciruits", QCQC '98, Springer-erlg, pp (1999) 7. Willims C. P., Clerwter S. H., "Explortions in Quntum Computing", Springer-erlg, New York In. (1998) 8. T. Yuki nd H. I. Geneti lgorithms nd quntum iruit design, evolving simpler teleporttion iruit, In Lte Breking Ppers t the 2000 Geneti nd Evolutionry Computtion Conferene, pp , G. Negoveti, M. Perkowski, M. Luk, A. Buller, Evolving quntum iruits nd n FPGA-sed Quntum Computing Emultor, Pro. Intern. Workshop on Boolen Prolems, Brssrd G., Brunstein S. L., Cleve R., Teleporttion s Quntum Computtion, in Proeedings of the Fourth Workshop on Physis nd Computtion, New Englnd Complex System Institute. 11. L.K. Grover, A Frmework for Fst Quntum Mehnil Algorithms, ACM Symposium on Theory of Computing (STOC), A. Breno et l., Elementry Gtes For Quntum Computtion, Physil Review A 52, 1995, pp M. Nielsen & I. Chung, Quntum Computtion nd Quntum Informtion, Cmridge Univ. Press, Septemer T. Hogg et l., Tools for Quntum Algorithms, J. Smolin, D. P. Diinenzo, Five two-quit gtes re suffiient to implement the quntum Fredkin gte. Physil Review A, ol. 53, no. 4, April 1996, pp M. Luk nd M. Perkowski, Evolving Quntum Ciruits Using Geneti Algorithm, Pro. of NASA/DOD Workshop on Evolvle Hrdwre, Wshington, D.C. July M. Luk, M. Pivtoriko, A. Mishhenko, nd M. Perkowski, Automted Synthesis of Generlized Reversile Csdes using Geneti Algorithms, Pro.5 th Intern Workshop on Boolen Prolems, Freierg, Germny, Septemer 19-20, pp Miller, D. M., Spetrl nd Two-Ple Deomposition Tehniques in Reversile Logi, Pro. Midwest Symposium on Ciruits nd Systems, on CD-ROM, August

14 D. M. Miller nd G.W. Duek, Spetrl Tehniques for Reversile Logi Synthesis, sumitted to RM M. H. A. Khn, nd M. Perkowski, Multi-Output ESOP Synthesis with Csdes of New Reversile Gte Fmily, sumitted to RM Mslov, D., nd G. W. Duek, Grge in Reversile Designs of Multiple- Output Funtions, sumitted to RM Duek, G. W., nd D. Mslov, Reversile Funtion Synthesis with Minimum Grge Outputs, sumitted to RM E. Fredkin nd T. Toffoli, Conservtive logi, Interntionl Journl of Theoretil Physis, 21, pp , P. Kerntopf, Synthesis of multipurpose reversile logi gtes, Proeedings of EUROMICRO Symposium on Digitl Systems Design, 2002, pp J-S. Lee, Y. Chung, J. Kim, nd S. Lee, A Prtil Method of Construting Quntum Comintionl Logi Ciruits, rxiv:qunt-ph/ v1, 12 Nov J. Kim, J-S Lee, nd S. Lee, Implementtion of the refined Deutsh-Jozs lgorithm on three-it NMR quntum omputer, Physil Review A, olume 62, , J. Kim, J-S. Lee, nd S. Lee, Implementing unitry opertors in quntum omputtion, Physil Review A, olume 62, , M. D. Prie, S.S. Somroo, A.E. Dunlop, T. F. Hvel, nd D. G. Cory, Generlized methods for the development of quntum logi gtes for n NMR quntum informtion proessor, Physil Review A, ol. 60, Numer 4, Otoer 1999, pp M.D. Prie, S.S. Somroo, C.H. Tseng, J.C. Core, A.H. Fhmy, T.F. Hvel nd D.G. Cory, Constrution nd Implementtion of NMR Quntum Logi Gtes for Two Spin Systems, Journl of Mgneti Resonne, 140, pp , A. Al-Rdi, L.W. Csperson, M. Perkowski nd X. Song, Multiple-lued Quntum Logi, Booklet of 11 th Post-Binry Ultr Lrge Sle Integrtion (ULSI) 2002 Workshop, pp , Boston, Msshusetts, 15 th My G. Yng, W.N.N. Hung, X. Song nd M. Perkowski, Mjority-Bsed Reversile Logi Gte, sumitted to RM D. Deutsh, Quntum omputtionl networks, Pro. Roy. So. Lond. A. 425, 1989, pp K. Iwm, Y. Kmyshi, nd S. Ymshit, Trnsformtion Rules for Designing CNOT-sed Quntum Ciruits, Pro. DAC 2002, New Orlens, Louisin. 34. P. Kerntopf, Mximlly effiient inry nd multi-vlued reversile gtes, Proeedings of ULSI Workshop, Wrsw, Polnd, My 2001, pp A. Khlopotine, M. Perkowski, nd P. Kerntopf, Reversile logi synthesis y gte omposition, Proeedings of IWLS pp A. Mishhenko nd M. Perkowski, Logi Synthesis of Reversile Wve Csdes, Pro. IEEE/ACM Interntionl Workshop on Logi Synthesis, June pp M. Perkowski, P. Kerntopf, A. Buller, M. Chrznowsk-Jeske, A. Mishhenko, X. Song, A. Al-Rdi, L. Jozwik, A. Coppol, B. Mssey, Regulrity nd symmetry s se for effiient reliztion of reversile logi iruits, Proeedings of IWLS 2001, pp , M. Perkowski, L. Jozwik, P. Kerntopf, A. Mishhenko, A. Al-Rdi, A.

15 Coppol, A. Buller, X. Song, M. M. H. A. Khn, S. Ynushkevih,. Shmerko, nd M. Chrznowsk-Jeske, A generl deomposition for reversile logi, Proeedings of RM pp M. Perkowski, P. Kerntopf, A. Buller, M. Chrznowsk-Jeske, A. Mishhenko, X. Song, A. Al-Rdi, L. Jozwik, A. Coppol, B. Mssey, Regulr reliztion of symmetri funtions using reversile logi, Proeedings of EUROMICRO Symposium on Digitl Systems Design, 2001, pp Shende, A.K. Prsd, I.L. Mrkov, J.P. Hyes, Reversile Logi Ciruit Synthesis, Pro. 11 th IEEE/ACM Intern. Workshop on Logi Synthesis, 2002, pp A. Al-Rdi, Novel Methods for Reversile Logi Synthesis nd Their Applition to Quntum Computing, Ph. D. Thesis, Portlnd Stte University, Portlnd, Oregon, USA, Otoer 24, M. Perkowski, J. Liu, nd J. Brown, "Quik Softwre Prototyping: CAD Design of Digitl CAD Algorithms," In G. Zorist, (ed.), "Progress in Computer Aided LSI Design," ol. 1., Alex Pulishing Corp., pp , M. Perkowski, M. Luk, et l, Tehnology mpping for Quntum CAD, in preprtion.