A Hierarchical Approach to Computer-Aided Design of Quantum Circuits

Transcription

1 A ierrhil Approh to Computer-Aided Design of Quntum Ciruits Mrek Perkowski,+* Mrtin Luk,* Mikhil Pivtoriko,* Pwel Kerntopf, & Mihele Folgheriter ^, Dongsoo Lee, + yungok Kim,+ oong wngo, Jung-wook Kim+ nd Yong oo 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, Guseong-dong, Yuseong-gu, Tejeon, , Repuli of Kore, & Institute of Computer Siene, rsw University of Tehnology, Nowowiejsk 15/19, rsw, Polnd, ^ Diprtimento di Elettroni ed Informzione (DEI), Politenio di Milno, Pizz Leonrdo d Vini 32, Milno, Itly. Astrt: A new pproh to synthesis of quntum logi iruits hs een presented 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). The pproh is sed on the following priniples: (1) design of (omplex) gtes using pproh tht links evolutionry, omintoril nd humn intertion pprohes, (2) design of iruits with these gtes using evolutionry nd serh methods, (3) mro-genertion of omplex gtes to lower level quntum primitives, (4) pplying tehnology-relted optimizing tutologil trnsforms tht tke relisti quntum lyout onstrints, (5) lulting relisti osts relted to quntum reliztion, (6) synthesis of inompletely speified funtions. 1. Introdution hile quntum mehnis nd quntum omputing re estlished reserh res, utomted quntum iruit synthesis is still only t the eginning of its explortion [9,30,45,48,49,50,51]. 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 two immedite pplition res: (1) Optimizing quntum iruits for NMR [19,20,21,22,42,43], 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). idth is solutely ritil nd length is lso importnt euse of deoherene. Ciruit redution is very importnt for present tehnology, so ext or suminiml 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 muh more diffiult. (2) Modeling quntum omputers in FPGA-sed reonfigurle hrdwre for speeding-up omputtions tht re very ineffiient on stndrd omputers [31]. Sine for urrent FPGA tehnologies only smll quntum iruits n e emulted or simulted, the requirements re similr to point (1). 2. Quntum iruit synthesis is not the sme s stndrd reversile logi synthesis The mjor differene etween quntum logi nd inry logi is the onept of the informtion itself. hile 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 [50]. 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

2 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 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 multiplition (tensor produt) opertion [59]. As n exmple, onsider two quits with ψ 1 =α 1 0>+β 1 1> nd ψ 2 =α 2 0>+β 2 1>. hen 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. 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 re eing developed. Proilisti lultions sed on this representtion re used in only very smll quntum omputers so fr, 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. Moreover it ws shown [50] tht ny quntum omputing hs to e reversile, whih ffets ll synthesis methods. 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. 3. Quntum CAD hierrhil pproh. Building 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. In this pper we fous on the synthesis of ritrry quntum iruits (nd quntum gtes in prtiulr) of smll size, less thn seven vriles. As speil se, we design iruits from lss of permuttion gtes whih hve unitry mtries eing permuttion mtries. Suh iruits orrespond to lssil reversile Boolen funtions, ut in ontrst to previous pprohes our methods del with inompletely rther thn ompletely speified Boolen funtions. The reder hs, however, to keep in mind tht our design methods re for generl quntum iruits, with ritrry unitry mtries. The presented methods hve een speilized to some lsses suh s f-ontrolledphse-shift gte iruits [22], nd generlized to multiple-vlued quntum logi nd mixed multiple-vlued logi [1,2,17,39,40,41]. hile most of synthesis ppers del with generl reversile logi [13,25-27,46] here we re interested in quntum reliztions. e wnt to demonstrte nd emphsize tht it is not true tht urrent inry reversile logi synthesis methods n e diretly pplied to permuttion quntum iruits. This wy we wnt to rete new improved methods tht will e more prtil for optimizing quntum sequenes in urrent existing tehnologies suh s NMR [19-22,42,43]. e thus show (see lso [52]) tht Toffoli (CCNOT) nd Fredkin re not lwys the est gtes for quntum omputing. Multi-input Toffoli gte looks simple in digrm, ut tkes lot of gtes when mro-generted to 3*3 gtes with uxiliry onstnts suh gtes re not primitives. Therefore we propose hierrhil pproh design of powerful gtes first. Synthesis should e performed on one hnd t lower level of quntum primitives suh s CNOT (Feynmn), ontrolled squre-root-of-not nd dmrd (or even lower), nd on the other hnd t the level of more powerful reversile gtes, suh s the omplex gtes introdued in [1,2,6,14-18,25,29,36-41,52]. Using diverse nd more omplex gtes simplifies serh. The trnsformtion from

3 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. For instne in NMR every gte must e onstruted from 2-quit gtes, ut some pirs of (quntum) wires of (quntum) rry nnot e inputs/outputs of gte nd thus swp gtes must e dded. One swp gtes osts three Feynmn gtes so it is ostly. A good synthesis method should then involve not only logi synthesis ut lso Quntum Lyout Synthesis, whih in this prtiulr se mens trnsforming iruit in suh wy tht gtes re not onneting the foridden pirs of quntum wires. This leds to prolems of gte ordering, input vrile ordering, lol trnsformtions, removl of swp gtes (plnriztion) nd others. This wy, the 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. ere we propose methods tht re tuned to rel tehnology. These optimiztion prolems re quite similr to tehnology mpping nd physil design res in lssil CAD. So fr, no systemti CAD pprohes hve een proposed for them nd they re only solved d ho y physiists who uild NMR omputers [22]. In our CAD tools, the evolutionry, omintoril serh, nd humn-oriented intertive methods re omined. e were le to synthesize ompletely utomtilly more omplex iruits thn those found y previous progrms, for instne the EPR iruits nd Toffoli or Mrgolus gtes. 4. New Quntum Gtes nd their Cost Funtions for the Optimiztion Algorithms Figure 1 Figure 2 = To V V V +.. 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 illustrte more preise ost funtions for gtes tht re used in our optimiztion lgorithms (even more urte osts n e found in [39,40]). ere, we follow the footsteps of previous ppers in quntum omputing [3,19-22,42,43,47] nd will relize ll gtes from 1* 1 nd 2*2 gtes. Moreover, ording to [47] we ssume tht the ost of every 2*2 gte is the sme. e 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 [47], the ost of Toffoli gte is five 2*2 gtes, or simply, 5. In Figure 1, V is squre-root-of-not gte (unitry mtrix V) nd V + is its hermitin. Thus V V retes unitry mtrix of NOT gte nd V V+ = 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 [3]). Now we will relize the Fredkin gte from the Toffoli gte. Using evolutionry synthesis methods disussed elow, we synthesize the Fredkin gte using two Feynmn nd one Toffoli gte s in Figure 2. 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 [47] 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, we lulted osts of other known gtes [39,40,52] nd me to similr

4 oservtions - the ost of Miller s Gte is 7 nd not 9, s might e expeted from its inry shemtis using Feynmn nd Toffoli gtes. Another reliztion of Miller s gte hs n even smller ost of 6. There re similr gtes of ost 5. Similrly the osts of 3*3 gtes y Kerntopf [14,15,36], Mrgolus [14], De Vos [15], Khn [16], nd Mslow [6,25], osts of ll 4*4 Perkowski s gtes [18], nd other gtes from [15] 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 investigtions. e found tht the eqution of this gte ws known to Peres [14], ut it hs een not used in reversile or quntum omputing. Oserve tht lgorithms from [13,46], when given the eqution of Peres gte, would return the solution omposed from Toffoli nd Feynmn gtes, whih would led to lerly non-miniml quntum sequene. owever, if the post-proessing stge of mro-genertion nd next equivlene-sed optimizing trnsformtions were pplied to the results from [13,46], then the minimum solution would e found. It is now hllenge for 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. [3]). Conluding, the synthesis of NMR iruits should e not restrited to Toffoli, Feynmn nd NOT gtes s in [13,25-27,46]. Figure 3 ) V V V + Figure 4 Cost = four 2*2 quntum gtes ) V V V + ) ) V V V + V V V + = (+) C=0 ==> (+) C=1 ==> (+) = 5. Frme-sed serh nd evolutionry gte genertor s Now oserve in Figure 4, tht the Peres Gte (Fig. 4) 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 V + 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 4. 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. There exist very mny omintoril possiilities of onneting Feynmn (nd other) gtes to inputs nd outputs of si gtes, s ws shown in setion 4. The ove illustrted proedure of systemti dding gtes to strutures hs een utomted [39,40]. Our frmesed 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 versus AND/OR, et), numer of node hildren, et. (reder interested in pseudo-ode nd detils is referred to [7,33-35]. 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 nmes of optimizing (omining) trnsformtions to e pplied, s well s the osts of gtes, re the prmeters of the progrm. These gtes re like slots in frme struture known from AI progrmming. The sequene of slots retes GA s hromosome. This new method, whih we ll informed GA, omines evolutionry progrmming nd frmes, s well s the user eing prt of the design loop. The detils out evolutionry lgorithms for oth truly quntum iruits nd permuttion iruits n e found in [23] nd [24], 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. Some gtes in this lirry re gered towrds strutured design methods (suh s Kerntopf gte used in

5 regulr strutures lled Nets nd other gtes used in Ltties [2,15,36,38]). All gtes hve good reliztions (short nd ompletely lyout-relizle sequenes) in given prtiulr quntum tehnology, suh s espeilly NMR. In ll our progrms we use similr enodings of quntum rrys. 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 5. It illustrtes trnsformtion of QC from the enoded hromosome (5) to finl quntum iruit nottion representtion of this QC (5). ere S is Swp gte, is dmrd gte, is wire. In Fig 5 there is one CCNOT (Toffoli) gte. In Figure 5 it is shown how the iruit from Figure 5 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. hile 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 distne s prt of fitness funtion lultion. The exmple shown in Figure 5 illustrtes ommon inonveniene of enoding quntum iruits for geneti lgorithms. A quntum gte CCNOT n e pled over three different ritrry quntum wires in quntum iruit. owever 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 5 illustrtes lso the Quntum Lyout prolem. Assuming tht quntum wires 2 nd 4 re nonrelizle pir, two swp gtes s shown in Figure 5 should e inserted to mke the iruit from Figure 5 relizle in this prtiulr NMR tehnology. The ost of the iruit should then inlude the totl ost of ll swp gtes, whih n e very high in terms of 2-quit gtes (see [16,17,39]). Figure 6 illustrtes how prllel nd seril loks orrespond to Kroneker produts nd mtrix produts * while ost (fitness) funtions re lulted this stge is slowdown of the progrms. 1 Figure 5 3 Figure 6 S S S S 2 ( ( ( Figure 7 () () ()

6 6. Synthesis, Mro-genertion nd Optimizing Trnsformtions. Synthesis of permuttion iruits n e lso performed using methods from [18,28,32,39,40]. 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 [13,46]) 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 mro-genertion [3]. These trnsformtions generlize ( ut re in priniple similr to) those presented in detil in [7,13,19-22,28,32,39,40,42,43,46] nd will e not disussed here. Oserve, however, tht our trnsformtions re more generl thn those from [13,46] euse: (1) they pply to inomplete funtions, (2) they exhust in sense spe of trnsformtions [28,32], (3) they re used for omplex gtes nd mro-genertion, (4) they pply lso to truly quntum (non-permuttive) gtes [27,28], (5) they pply to generl Glois Logi nd not only to GF(2) [40]. The ontrol of the trnsformtions is sed on methods from [7,33-35], thus the user n experiment with vrious trnsformtion strtegies. Mny of them re sed on ovious rules of EXOR lger, suh s some illustrted ove, ut other use Glois Field or Lee lgers. For instne, it n e shown how Unified Complex dmrd Trnsform of Rhrdj nd Flkowski [44], not investigted so fr in quntum omputing, n e optimized using non-permuttion dmrd gtes, other gtes nd quntum primitives from [20,42]. 7. 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 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. owever 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 [45, 49]. e seleted from literture three enhmrks shown in Figure 7. The first two re oth iruits to produe EPR s in [45], the lst is the send iruit originlly proposed y [4] nd evolved in [49]. The results re shown elow in Tle 2. e 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.

7 Numer of inputs per q- gte Numer of genertions 3 - input < inputs <350 pm pc Rel time (verge 20 runs) pm<0.2 Numer of genertions Rel time (verge 20 runs) < 1 minutes <300 < 2 minute < 2 minutes <900 < 3 minute Popultion size 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. Using this lgorithm, we were not le to find less expensive quntum reliztions of ny 3*3 permuttion gtes thn Smolin-like 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 illims, 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. 8. Conlusion In ontrst to previous pulished work, we produe sequene of quntum opertions tht is prtilly relizle. Our hierrhil stges n e ompred to stndrd gte lirry design, generi logi synthesis nd tehnology mpping stges of lssil CAD systems, respetively. The designer n e 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. e hve shown tht the hierrhil pproh n e used for utomted QC synthesis. Our evolutionry progrm found one new iruit tht ws erlier me ross y illims [507], three iruits loted y Ruinstein [45] 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 [50] nd entnglement [45] our pproh is fully generl. It is lso hierrhil nd llows to inlude humns in the design loop. The lgorithm from [23] is pplied to permuttion iruits suh s those used in the fmous Grover s Quntum Serh Algorithm [11]. 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 to reversile nd quntum iruit synthesis. e reted enhmrk lirry to e used y other interested QC reserhers: 9. Referenes 1. A. Al-Rdi, L.. Csperson, M. Perkowski nd X. Song, Multiple-Vlued Quntum Logi, Booklet of 11 th Post-Binry Ultr Lrge Sle Integrtion (ULSI) 2002 orkshop, pp , Boston, Msshusetts, 15 th My 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, A. Breno et l., Elementry Gtes For Quntum Computtion, Phys. Review A 52, 1995, pp G. Brssrd., S.L. Brunstein, nd R. Cleve, Teleporttion s Quntum Computtion, in Pro. 4 th orkshop on Physis nd Computtion, New Englnd Complex System Institute. 5. D. Deutsh, Quntum omputtionl networks, Pro. Roy. So. Lond. A. 425, 1989, pp G.. Duek, nd D. Mslov, Reversile Funtion Synthesis with Minimum Grge Outputs, Pro. RM K.Dill nd M. Perkowski, The Cretion of Cyerneti (Multi-Strtegi Lerning) Prolem-Solver: Automtilly Designed Algorithms for Logi Synthesis nd Minimiztion, sumitted to journl, E. Fredkin nd T. Toffoli, Conservtive logi, Intern. J. Theoretil Physis, 21, pp , Y. Z. Ge, L. T. tson, nd E. G. Collins. Geneti lgorithms for optimiztion on quntum omputer. In

8 Unonventionl Models of Computtion, pp D. E. Golderg, Geneti Algorithms in Serh, Optimiztion, nd Mhine Lerning Addison esley, L.K. Grover, A Frmework for Fst Quntum Mehnil Algorithms, Pro. STOC, T. ogg et l., Tools for Quntum Algorithms, K. Iwm, Y. Kmyshi, nd S. Ymshit, Trnsformtion Rules for Designing CNOT-sed Quntum Ciruits, Pro. DAC 2002, New Orlens, Louisin. 14. P. Kerntopf, Mximlly effiient inry nd multi-vlued reversile gtes, Pro. ULSI orkshop, rsw, Polnd, My 2001, pp P. Kerntopf, Synthesis of multipurpose reversile logi gtes, Proeedings of EUROMICRO Symposium on Digitl Systems Design, 2002, pp M.. A. Khn, nd M. Perkowski, Multi-Output ESOP Synthesis with Csdes of New Reversile Gte Fmily, Pro. RM M..A. Khn, nd M. Perkowski, Multi-Output Glois Field Sum of Produts (GFSOP) Synthesis with New Quntum Csdes, sumitted to ISMVL A. Khlopotine, M. Perkowski, nd P. Kerntopf, Reversile logi synthesis y gte omposition, Pro. ILS pp J. Kim, J-S Lee, nd S. Lee, Implementtion of the refined Deutsh-Jozs lgorithm on three-it NMR quntum omputer, Physil Review A, Volume 62, , J. Kim, J-S. Lee, nd S. Lee, Implementing unitry opertors in quntum omputtion, Physil Review A, Volume 62, , J. Kim, A Study on Ensemle Quntum Computers, Ph.D. Thesis, Kore Advned Institute of Siene nd Tehnology, My 23, 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 M. Luk nd M. Perkowski, Evolving Quntum Ciruits Using Geneti Algorithm, Pro. of NASA/DOD orkshop on Evolvle rdwre, shington, D.C. July2002, pp M. Luk, M. Pivtoriko, A. Mishhenko, nd M. Perkowski, Automted Synthesis of Generlized Reversile Csdes using Geneti Algorithms, Pro.5 th Intern orkshop on Boolen Prolems, Freierg, Germny, Septemer 19-20, pp D. Mslov, nd G.. Duek, Grge in Reversile Designs of Multiple-Output Funtions, Pro. RM D.M. Miller, Spetrl nd Two-Ple Deomposition Tehniques in Reversile Logi, Pro. Midwest Symposium on Ciruits nd Systems, on CD-ROM, August D. M. Miller nd G.. Duek, Spetrl Tehniques for Reversile Logi Synthesis, Pro. RM A. Mishhenko nd M. Perkowski, ``Fst euristi Minimiztion of Exlusive Sums-of-Produts,'' Pro. RM'2001 orkshop, August A. Mishhenko nd M. Perkowski, Logi Synthesis of Reversile ve Csdes, Pro. ILS, June pp M. Nielsen & I. Chung, Quntum Computtion nd Quntum Informtion, Cmridge Univ. Press, Septemer G. Negoveti, M. Perkowski, M. Luk, A. Buller, Evolving quntum iruits nd n FPGA-sed Quntum Computing Emultor, Pro. Intern. orkshop on Boolen Prolems, pp N. Song, nd M. Perkowski, Minimiztion of Exlusive Sums of Multi-Vlued Complex Terms for Logi Cell Arrys, Pro. ISMVL 98, Fukuok, Jpn, My M. Perkowski, "The method of solving omintoril prolems in the utomti design of digitl systems". Ph.D. Thesis, Institute of Automti Control, Tehnil University of rsw, M. A. Perkowski, P. Dysko, B. J. Flkowski, "Two Lerning Methods for Tree-Serh Comintoril Optimizer," Pro. of IEEE Interntionl Phoenix Conferene on Computers nd Communition, pp , Sottsdle, Arizon, Mrh M. Perkowski, J. Liu, nd J. Brown, "Quik Softwre Prototyping: CAD Design of Digitl CAD Algorithms," In G. Zorist, (ed.), "Progress in Computer Aided VLSI Design," Vol. 1., Alex Pulishing Corp., pp , M. Perkowski, P. Kerntopf, A. Buller, M. Chrznowsk-Jeske, A. Mishhenko, X. Song, A. Al-Rdi, L. Jozwik, A. Coppol, nd B. Mssey, Regulrity nd symmetry s se for effiient reliztion of reversile logi iruits, Proeedings of ILS 2001, pp , M. Perkowski, L. Jozwik, P. Kerntopf, A. Mishhenko, A. Al-Rdi, A. Coppol, A. Buller, X. Song, M. M.. A. Khn, S. Ynushkevih, V. 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, nd B. Mssey, Regulr reliztion of symmetri funtions using reversile logi,

9 Proeedings of EUROMICRO Symposium on Digitl Systems Design, 2001, pp M. Perkowski, P. Kerntopf, A. Al-Rdi, nd M..A. Khn, Multiple-Vlued Quntum Computing. Issues, Open Prolems, Solutions, Tehnil Report, KAIST, Deemer M. Perkowski, M. Pivtoriko, M. Luk, C. ung, S. Ahmd, et l, Tehnology mpping for Quntum CAD, in preprtion. 41. M. Perkowski, A. Al-Rdi, P. Kerntopf, Multiple-Vlued Quntum Logi Synthesis, Pro Intern. Symp. on New Prdigms VLSI Computing, Deemer 12-14, Sendi, Jpn, pp M. D. Prie, S.S. Somroo, A.E. Dunlop, T. F. vel, nd D. G. Cory, Generlized methods for the development of quntum logi gtes for n NMR quntum informtion proessor, Physil Review A, Vol. 60, Numer 4, Otoer 1999, pp M.D. Prie, S.S. Somroo, C.. Tseng, J.C. Core, A.. Fhmy, T.F. vel nd D.G. Cory, Constrution nd Implementtion of NMR Quntum Logi Gtes for Two Spin Systems, Journl of Mgneti Resonne, 140, pp , S. Rhrdj nd B.J. Flkowski, Fmily of Unified Complex dmrd Trnsforms, IEEE Trns. Ciruits nd Systems II: Anlog nd Digitl Signl Proessing, Vol. 46, No. 8, pp , B.I.P. Ruinstein, Evolving quntum iruits using geneti progrmming, Pro. CEC2001, pp V.V. Shende, A.K. Prsd, I.L. Mrkov, J.P. yes, Reversile Logi Ciruit Synthesis, Pro. ILS, 2002, pp J. Smolin, D. P. DiVinenzo, Five two-quit gtes re suffiient to implement the quntum Fredkin gte. Physil Review A, Vol. 53, no. 4, April 1996, pp L. Spetor,. Brnum,. J. Bernstein, nd N. Swmy, Finding etter-thn-lssil quntum AND/OR lgorithm using geneti progrmming, In Pro Congress on Evolutionry Computtion, volume 3, pp , shington D.C., July 1999, IEEE, Pistwy, NJ. 49. C.P. illims, A.G. Gry, "Automted Design of Quntum Ciruits", QCQC '98, Springer-Verlg, pp , C.P. illims, S.. Clerwter "Explortions in Quntum Computing", Springer-Verlg, N Y T. Yuki nd. I. Geneti lgorithms nd quntum iruit design, evolving simpler teleporttion iruit, In Lte Breking Ppers t the 2000 Geneti nd Evolutionry Comp. Conf. pp , G. Yng,.N.N. ung, X. Song nd M. Perkowski, Mjority-Bsed Reversile Logi Gte, Pro. RM 2003.