Synthesis of Quantum Circuits in Linear Nearest Neighbor Model Using Positive Davio Lattices

Transcription

1 Portln Stte University PDXSholr Eletril n Computer Engineering Fulty Pulitions n Presenttions Eletril n Computer Engineering 4-20 Synthesis of Quntum Ciruits in Liner Nerest Neighor Moel Using Positive Dvio Ltties Mrek Perkowski Portln Stte University, mrek.perkowski@px.eu Mrtin Luk Dipl Shh Portln Stte University Mihitk Kmeym Let us know how ess to this oument enefits you. Follow this n itionl works t: Prt of the Eletril n Computer Engineering Commons Cittion Detils Perkowski, Mrek; Luk, Mrtin; Shh, Dipl; n Kmeym, Mihitk, "Synthesis of Quntum Ciruits in Liner Nerest Neighor Moel Using Positive Dvio Ltties" (20). Eletril n Computer Engineering Fulty Pulitions n Presenttions This Artile is rought to you for free n open ess. It hs een epte for inlusion in Eletril n Computer Engineering Fulty Pulitions n Presenttions y n uthorize ministrtor of PDXSholr. For more informtion, plese ontt pxsholr@px.eu.

2 FACTA UNIERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 24, no., April 20, 7-87 Synthesis of quntum iruits in Liner Nerest Neighor moel using Positive Dvio Ltties Mrek Perkowski, Mrtin Luk, Dipl Shh, n Mihitk Kmeym Astrt: We present logi synthesis metho se on ltties tht relize quntum rrys in One-Dimensionl Ion Trp tehnology. This mens tht ll gtes re uilt from 2x2 quntum primitives tht re lote only on neighor quits in oneimensionl spe (lle lso vetor of quits or Liner Nerest Neighor (LNN) rhiteture). The Logi iruits esigne y the propose metho re relize only with 3*3 Toffoli, Feynmn n NOT quntum gtes n the usge of the ommonly use multi-input Toffoli gtes is voie. This reliztion metho of quntum iruits is ifferent from most of reversile iruits synthesis methos from the literture tht use only high level quntum ost se on the numer of quntum gtes. Our synthesis pproh pplies to oth stnr n LNN quntum ost moels. It les to entirely new CAD lgorithms for iruit synthesis n sustntilly ereses the quntum ost for LNN quntum iruits. The rwk of synthesizing iruits in the presente LNN rhiteture is the ition of nill quits. Keywors: Reversile logi synthesis, lttie, leiner nerest neighor moel. Introution: Stnr versus Liner Nerest Neighor quntum ost moels Most ppers in the literture out utomte synthesis of quntum n reversile (permuttive) iruits re not relte to ny prtiulr quntum reliztion tehnology [, 2, 3, 4, 5, 6]. Their moels ssume tht gte n e relize on ny suset of quits. The moel use in most of the previous permuttive quntum Mnusript reeive on Ferury 2, 20. M. Perkowski n D. Shh re with Deprtment of Eletril n Computer Engineering, Portln Stte University, Portln, OR, USA, (emil: mperkows@ee.px.eu). M. Luk n M. Kmeym re with Grute Shool of Informtion Sienes, Tohoku University, Seni, Jpn, (emil: lukm@kmeym.eei.tohoku..jp). 7

3 72 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: iruit synthesis ppers ssumes tht there n exist gte lote etween ny two quits, even if these quits re lote fr wy in physil spe (in vetor) one from nother. This (very pproximte) ssumption my e suffiient to lulte quntum osts for very smll iruits. This ssumption ws epte in theoretil frmework ut from prtil point of view n with respet to prtiulr tehnologies (suh s Ion trp in this se) reting gtes on ritrry quits is not only extremely iffiult ut lso ost ineffetive; eh gte hs to e properly onverte n relize in n LNN rhiteture. Thus, in generl rhiteture inepenent synthesis moels re suffiient to pproximte the rel ost of smll iruits. For lrger quntum iruits relize in the future s well s for urrently relizle iruits with out 2 quits rhiteture epenent ost moels n synthesis methos re require. For instne in quntum optis [7, 8] suh rhiteturl moels require more evelopment to tke into ount more omplex onstrints suh s time propgtion n physil size. There exists no single tehnology for whih this moel is vli. In ontrst, for vrious reliztion tehnologies there exist ifferent neighorhoos of quits [9, 0]. For instne, in the One-Dimensionl Ion Trp tehnology [] the quits rete liner, one-imensionl (D) vetor, the Liner Nerest Neighor (LNN) moel (rhiteture). In quntum optis, quits lso intert y proximity using optil wires or rystls [2, 3, 7]. Therefore, it is sfe to ssume tht the LNN ost moel is urrently one of the most pproprite moels for urrent tehnologies. Ciruits relize in LNN use quntum gtes efine only on neighor quits n the gtes re uilt from x n 2x2 quntum primitives. We elieve tht LNN moel shoul e use for Ion Trp n similr tehnologies n new quntum ost moels shoul e evelope for other speifi tehnologies. With respet to generl quntum iruits the LNN Moel ws introue y Fowler et l [4] for esigning Quntum Fourier trnsform iruit. Their work ws improve in [5]. Pper [0] onsiers theoretil spets of tehniques for trnslting quntum iruits etween vrious rhitetures. The first pper out permuttive quntum iruits esign with the LNN moel ws written y Curo et l [6] n they esigne ripple-rry ition iruit. Automte synthesis of generl quntum iruits with LNN moel ws first introue riefly in [7] ut no speifi metho ws presente n results nlyze. Chkrrti n Sur- Koly [8, 9] presente nlysis of osts of single-output FPRM-se reversile iruits. Methos for generl quntum iruit for the LNN moel were isusse y Hirt et l [20] n other uthors [2, 22, 23, 24] rete vrious methos to synthesize reversile quntum iruits in the LNN moel. These methos re lle nerest neighor quntum synthesis. For instne, Hirt s metho [20] strts from n ritrry quntum rry n moifies it to the LNN Arhiteture y

4 Synthesis of quntum iruits in Liner Nerest Neighor moel inserting SWAP gtes n minimizing their numer. The vntge of this metho is tht it n e pplie to n ritrry quntum iruit. The numer of e gtes is however exessive n the properties of permuttive iruits eing the speil se of quntum iruits re not tken into ount. Thus eveloping n lgorithm speifilly esigne for the LNN rhiteture n signifintly improve the ost of iruits relize in the LNN moel euse the iruits re speifilly rfte to mth the rhiteture rther thn esigning reversile iruits with ritrry gtes n then moifying it to mth the rhiteture. Moreover, most of the methos whether using the LNN moel or not o not evlute n ompre the ifferenes etween the use ost n the LNN moel. This mens tht espite liming miniml results while using multi-ontrolle Toffoli (MCT) gtes the sme results n e shown to e non optiml when using the LNN moel. We present here new pproh se on ltties, whih pplies to only permuttive (reversile quntum) iruits. The metho oes not only exploit the lol minimiztion of SWAP gtes s in the previous works suh s [20, 24] ut, y synthesizing iruits s lttie the metho uses the lttie struture to esign iruits tht re less ostly when esigne for the LNN moel. We strt from n ritrry (non-reversile) Boolen funtion n relize it s reversile quntum iruit; the metho presente here onverts non-reversile funtion to reversile iruit y ing nill quits. The propose pproh presents for the first time onversion of n ritrry Boolen funtion to iruit with quntum ost moel tht tkes tehnology-relte onsiertions into ount in the logi synthesis lgorithm. It uses two quntum ost funtions; stnr quntum ost n Liner Nerest Neighor moel (LNN). The LNN moel ssumes tht iruit is esigne or moifie in suh wy tht it is ompose of only * gtes n 2*2 gtes on neighoring quits. To evlute our pproh we ompre the two osts methos s well s we ompre the otine osts with vrious previous lgorithms. The min ontriutions of the propose lgorithm is in the ft tht it llows to generte reversile logi iruits in the LNN moel with prtilly hieve minimum of SWAP gtes. This is verifie with other lgorithms the generte iruits with higher numer of not only SWAP gtes ut higher numer of gtes in generl. The pper is orgnize s follows. Setion 2 explins the stnr n LNN moels of lulting quntum osts. Setion 3 presents how one type of the previously introue Lttie Digrms, the Positive Dvio Ltties, n e pte to regulr reliztions of quntum iruits for the stnr n LNN moel moels. Setion 4 presents our experimentl results with oth ost moels n Se-

5 74 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: tion 5 onlues the pper. The pper ssumes tht the reer is fmilir with si quntum gtes n reversile logi onepts n with previous works on lttie lgorithms. 2 Motivtion for the LNN moel for quntum rrys relize in Ion Trp A gte etween ny two quits woul men n immeite iret intertion etween ny two ions in the Ion Trp, whih is physilly impossile in this tehnology ue to spe seprtion [, 25]. In the simplest (ut prtil s of 20) se, ll ions in Ion Trp re ple linerly (s One-Dimensionl vetor). Every ion (quit) n intert with t most one neighor ove n one neighor elow. This physil onstrint of 2-neighor quntum lyout of the sustrte hs muh influene on prtil esigns. As n exmple of prolems with LNN iruit moel, onsier the very simple 4x4 Toffoli gte shown s unit in Fig. (). Other uthors [26, 27, 28, 29, 30, 3, 32] lulte the quntum ost of the gte s funtion of numer of inputs regrless of wht is the istne of the quits use in this gte. This is not urte when the iruit is relize in liner Ion Trp tehnology. Nor is it goo for quntum optis or NMR tehnology tht is urrently in use. To relize this iruit in the LNN moel, one nill it shoul e e s in Fig. (). Next, eh of the 3x3 (stnr) Toffoli gtes from Fig. () re mro-generte to the Breno s reliztion of this gte [33], thus reting the quntum rry in Fig. (). This woul e fine if every two quits n intert iretly: ut they nnot. So trnsformtions from Fig. () to rete 2-neighor-only type of iruits re require. The finl iruit for the gte from Fig. () is then shown in Fig. (e). It hs 27 2x2 gtes in 2-neighors-only topology fter the minimiztion of ertin gtes. There re other wys to relize this gte in lyout, even without nill it. They re however even more expensive when relize in liner Ion Trp. The numer woul e even higher if the gte woul e relize on five quits tht re not neighors. Bse on the ove exmple, the quntum iruits in the LNN rhiteture shoul hve short onnetions insie gtes. As isusse in [34, 35, 36] short onnetions require regulr strutures suh s Ltties [34, 37, 38, 39] rete y pttion n generliztion of Akers Arrys [40]. The metho propose here uses Positive Dvio Ltties (PDL) [4, 42]. The reson for using PDL omes from the ft tht fter we nlyze the mpping of Lttie igrms to LNN rhiteture iruits, we foun tht the internl onnetions of the Lttie n e mppe well, i.e. with smll istnes. There is however ig troule with onneting ll Toffoli gtes to input vriles: this involves very mny SWAP gtes. This is illustrte

6 Synthesis of quntum iruits in Liner Nerest Neighor moel () () 0 () () 0 Fig.. This exmple illustrtes the nture of prolem with liner Ion Trp. A 4x4 Toffoli gte tht looks hep gte whih is however quite expensive when mppe to liner-neighorhoo quntum rry. () symol of gte s use y other uthors, () eompose Toffoli gte, () the finl iruit with 2-quit quntum primitives, ut not-relizle in liner neighorhoo s it hs wires going over gtes, () steps to relize the gte with wire going over it, (e) the finl iruit in liner neighorhoo Ion Trp. () in Fig. 4 n Fig. 5. Fig. 4 shows stnr quntum rry with nill it for funtion FX2 relize on PLA-like struture using only two-ontrolle quits Toffoli gtes n Feynmn Gtes. Fig. 3 shows the sme funtion rewritten to our D neighorhoo moel y ing SWAP gtes. This exmple illustrtes the ig ost of SWAP gtes when they re e to lulte relisti quntum ost for LNN moel of quits require in Ion Trp. The sme property n e shown on ny pulishe iruit for well-known enhmrks. Finlly, one n oserve tht the numer of the SWAP gtes require for n ritrry reversile iruit e mppe to the LNN moel n e pproximte nlytilly. Lemm 2.. An ritrry multi-ontrolle Toffoli (MCT) gte with k- ontrol its n trget it (together hving k its) tht is efine over p wires (p k, inluing

7 76 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: skippe wires) requires t mximum:%n relize in using the CNOT, C/C gtes requires t minimum ŝ = 2 (p k) (k )+(p ) 2 () with 2 (p k) (k ) representing the numer of swp gtes require to ring the ontrol its to the LNN proximity of the trget it n the (p ) 2 term represents the numer of SWAP gtes require to ring the quits insie of the Toffoli gte itself to the LNN neighorhoo. This mens tht for reversile iruit relize y only Toffoli gtes mximum of ŝ gtes will mke the iruit into LNN omptile iruit. Note tht eqution oes only speify how mny SWAP gtes re require to group the ontrols in LNN moel. Proof. Consier the Toffoli gte shown in Figure 2(). The gte is efine over 6 quits n hs 2 ontrol its n one trget. In this prtiulr se, the istne of the ontrol quits n of the trget is mximl n thus using formul from eq. we otin the orret result 2 SWAP gtes for outsie of the MCT gte (Figure 2()) n 4 s the totl numer of SWAP gtes(figure 2()). 2 (p k) (k ) () () () 2 (p k) (k ) + (p i) 2 Fig. 2. MCT gte efine over 6 quits. () relize s stnr MCT, () relize in the LNN moel, () relize in the LNN moel lso within the gte itself. Beuse the gte in Figure 2() hs mximl istne etween the ontrol n the trget quits, ny other onfigurtion of the sme gte will require less or equl to 4 SWAP gtes. Now let s look t more omplex exmple with two MCT gtes. The iruit in Figure 3() shows two MCT gtes onnete in series. Oserve tht when the iruit is uilt using stnr methos of synthesis (Figures 3()) - y uiling the iruit from MCT gtes n then onverting it to the LNN moel - the ost of the

8 Synthesis of quntum iruits in Liner Nerest Neighor moel SWAP gtes is muh higher thn when the iruit is uilt using synthesis methos for the LNN rhiteture (Figure 3()). This is euse in the lgorithm tht uils iruits for the LNN moel one n iretly preit whih lines shoul e returne to their initil position right fter eing use n whih not. () () () Fig. 3. Two MCT gtes efine over 6 quits. () relize s stnr MCT, () relize s two MCT gtes trnsforme to the LNN moel, () relize s two MCT esigne for the LNN moel. Finlly, we elieve tht LNN ost moel shoul reple the stnr ost moel for Ion Trp tehnology. New ost moels shoul e lso rete long these lines lso for other quntum tehnologies, rther thn using generl ost whih hs no reltion to ny tehnology tht we re wre of. Even for smll iruits osts lulte with the LNN moel iffer muh from stnr quntum osts n ifferent types of quntum synthesis metho show etter ost minimiztion ilities. Our new CAD tool for stnr n LNN moels of quntum osts is lle QULASYN (QUntum LAtties SYNthesizer). 0 grge grge grge grge 0 grge funtion Fig. 4. Ciruit for funtion FX2(,,,) = rete with our metho for tritionl quntum ost funtion lultion tht oes not tke Ion Trp tehnology onstrints into ount.

9 78 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: 3 Lttie Digrms with vrious types of expnsion gtes n their mpping to LNN moel As lrey introue, in this pper we pt Positive Dvio Lttie Digrms to quntum iruits [36]. Unlike in the stnr Shnnon Lttie Digrms tht uses multiplexers we restrit ourselves to uil quntum lttie equivlents for only Positive Dvio Ltties, using only 3*3 Toffoli, Feynmn n NOT gtes. 0 0 Fig. 5. Ciruit from Figure 4 moifie with ing SWAP gtes for new ost funtion lultion tht oes tke Ion Trp tehnology onstrints into ount, with 36 SWAP gtes e. It hs 36 SWAP gtes e to relize LNN moel quntum ost, oviously inrese. As n exmple, onsier the lssil Positive Dvio Lttie for funtion FX2(,,,) illustrte y igrm shown in Fig. 6. It is esigne using softwre presente in [37, 36]. The lgorithm for the lttie strts from logi Exlusive-Sum-of- Prouts (ESOP) eqution esriing the esire funtion. Initilly vrile is selete n oth the f x, f x n f x f x is lulte. This is repete for every vrile until ll ville vriles re onstnt. During the proess of the lttie onstrution reunnt noes re remove, merge with the gol of minimizing the size of the lttie. Suh lttie then n e further explore using for instne sifting or vrile repetition to otin the most esirle lttie. The Positive Dvio Lttie is next trnsforme to stnr form of quntum rry. For instne, to help the reer, the lttie from funtion F3(,,) is presente in Fig. 7() in form tht is intermeite etween Lttie Digrm n Quntum Arry. This intermeiry form is trnsforme s in Fig. 7(), where every intersetion of wires from Fig. 7() is reple y SWAP gte in Fig. 7(). This wy, new type of regulr struture relize in quntum rry with regulr onnetions is otine n the long onnetions typil for stnr Toffoli gtes re voie. Figure 7 illustrtes the essene of our trnsformtion metho from lttie igrms to quntum rrys of permuttive iruits. It explins lso why we use non-stnr nottion for intermeite stges. The whole trik ws to twist the igrm n reple intersetions of lines with SWAP gtes. This grphil

10 Synthesis of quntum iruits in Liner Nerest Neighor moel Fig. 6. Exmple of Positive Dvio Lttie from [37]. Positive Dvio Expnsion is pplie in eh noe. rile is repete. metho explins lso our SWAP insertion lgorithm.. The numer of SWAP gtes in our metho is however smller euse of regulrity of the new struture from Fig. 7(). We o not present here the etile lgorithms to rete Positive Dvio Ltties s they re isusse in full etil in previous ppers [37, 35, 36]. ut we provie high level esription for the ske of reer s unerstning. As n e seen the propose metho genertes itionl quits. The numer of the grge quits is the sme s in stnr quntum rry however there re itionl SWAP gtes require to relize the LNN moel. This results in iruits where there re no Toffoli gtes relize on non-neighor quits. The ost of e SWAP gtes is reltively low s eh suh gte n e relize with 3 Feynmn gtes [9], or EM pulses [43] fter optimiztion. Fig. 8 presents the trnsformtion of stnr Positive Dvio Lttie from Fig. 7() rwn in nother wy to regulr quntum rry with ition of SWAP gtes. Fig. 8 shows the trnsformtion from the mro-level to the CNOT/C gtes s well s the trnsformtion to the losest-neighor moel pplile for Ion trp tehnology. These Controlle-Squre-Root-of-Not (C) gtes n their hermitins re expline in etil in [9]. They re goo pproximtion of the quntum ost in the Eletromgneti (EM) pulses. Using the trnsformtions to pulses s shown in Fig. 9, the finl iruit ost n e lulte s follows. Eh of the loks of gtes shown in Fig. 8 hs the ost shown in Fig. 9(). The ost is 20x + 8x2 = 36. Beuse in etween the loks some of the gtes n e omine, it n e shown tht when the iruit

11 80 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: f grge grge () () f grge grge grge Fig. 7. Trnsformtion of funtion F3(,,) from lssil positive Dvio Lttie to Quntum Arry with Toffoli n SWAP gtes. Eh SWAP gte is next reple with 3 Feynmn gtes. () intermeite form, () finl Quntum Arry. Fig. 8. Trnsformtion of the iruit relize in Fig. 7 using Toffoli gte. Eh Toffoli n SWAP gtes re reple y quntum CNOT n C/C quntum gtes n rerrnge to stisfy the neighorhoo requirements of Ion trp.

12 Synthesis of quntum iruits in Liner Nerest Neighor moel... 8 from Fig. 8 is uilt in EM pulses its totl ost is 36x6-3 = 23. This wy, we n use the regulrity of quntum rry on the lowest implementtion level (quntum rottions level [9]) to further reue the numer of EM pulses. We use the metho from[43] to reue the numer of EM pulses. Fig. 9. The trnsformtions of loks of quntum gtes to the pulses level. The iruits in this pper re esigne using CAD tool QULASYN (QUntum LAtties SYNthesizer) tht uses oth the stnr n LNN moels of lulting quntum ost funtions. More etils on the implementtion of the QULASYN, on the vrile orering n repetition s well s on synthesis with new gtes n e foun in [36, 44]. The lgorithm uses vrious optimiztion tehniques n elertions in orer to perform n effiient KFDD suh s preorer serh, swpping of Dvio noes, sifting or exploittion of the symmetry of the KFDD. However these tehniques re not esrie in this pper s they re not proper to the KFDD pplie to reversile synthesis. The prtiulrity of the propose pproh is the ft of using only Toffoli, CNOT n the Not gte.

13 82 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: 4 Experimentl results Experimentl results for lulting Quntum rrys with tritionl quntum osts one y our QULASYN tool re given in Tle. We ompre our Lttie tool with MMD n Agrwl/Jh softwre. MMD stns for Miller, Mslov n Duek s lgorithm n AJ stns for Agrwl n Jh lgorithm. This tle shows vntge of pplying lttie se quntum synthesis even for tritionl ost funtions with gtes in stnr quntum rry. The est quntum osts re ol, itli n unerline. The sme osts for more thn one metho re ol n itli. Our tool rete the est result in 9 ses n in 8 ses the sme ost results were foun. Dshes re for results tht we hve no ess to. Tle. The trnsformtions of loks of quntum gtes to the pulses level. Benhmrk #Rel #Grge #Gres Cost CPU time #Gtes Cost #Gtes Cost inputs inputs Lttie Lttie Lttie DMM DMM AJ AJ 2to r < r < r < iter < iter < sym NA NA 9sym NA NA 5mo < mo < hm < hm < hm xor < xor < xnor < o < yle yle gryoe < gryoe < gryoe < nth prime3 in < nth prime4 in < nth prime5 in nth prime6 in Alu < hw < hw hw hw pprm < pprm pprm

14 Synthesis of quntum iruits in Liner Nerest Neighor moel Tle 2. The trnsformtions of loks of quntum gtes to the pulses level. Benhmrk #Gtes Cost #Gtes with Cost with #Gtes Cost #Gtes with Cost with Lttie Lttie SWAP SWAP gtes DMM DMM SWAP SWAP gtes insertion for Lttie insertion for MMD for Lttie for MMD 2to r r r iter iter sym sym mo mo hm hm hm Xor Xor Xnor eo Cyle Cyle Gryoe Gryoe Gryoe Nth prime3in Nth prime4in Nth prime5in Nth prime6in Alu hw hw hw hw pprm pprm pprm Tle 2 ompres QULASYN with other methos for the LNN Moel. Column is the nme of enhmrk, olumn 2 is the numer of gtes lulte for the stnr moel, n olumn 3 is the quntum ost for the stnr moel. Column 4 is the numer of gtes fter insertion of SWAP gtes to the lttie iruit. Column 5 is the respetive quntum ost with SWAP gtes inserte to lttie. Columns 6 to 9 give respetive results for MMD. The results for MMD metho were relulte y inserting the neessry SWAP gtes (lgorithms to insert SWAP gtes re given in [20, 8] n other ppers). To ompre thus quntum osts of Lttie metho with MMD one hs to ompre olumns 5 n 9. Bol itli numers shoul help

15 84 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: the omprison.the new metho is etter in 4 ses, n worse in 3 ses. In 4 ses the quntum osts re the sme. In some funtions like Hm7, r53, n hw6 the improvements of our metho for LNN moel ost re rmti. The reson tht moifie MMD is etter in some instnes is perhps use on the ft tht our tool is not fining the optiml orer of vriles in lttie, ut this shoul e n re of further reserh. 5 Conlusions We presente new synthesis metho of permuttive quntum iruits with two quntum ost funtions: stnr n LNN moel. Tles n 2 emonstrte strong improvements tht re rought y our metho in oth vrints. It shoul e however rememere tht our metho inreses the numer of nill quits, so the sme ritiism n pply to it s to other lgorithms tht introue nill quits. The numers of these nill quits n e foun in Tle. We o not lim in this pper to reple the stnr quntum osts with the LNN moel, we vote only to rete CAD tools tht will use severl tehnology-relte quntum osts. One of the most interesting spets of the presente pproh is the nturl onsequene of reuing the numer of SWAP gtes y simply mpping the reversile iruit on lttie. This mens tht simply representing the reversile iruit in the lttie hs for onsequene of mpping the iruit in the physil spe in suh mnner tht optimizes the reversile lyout with respet to the LNN moel. This pproh will e more explore in the future extensions of this work with respet to the presente lttie s well s with respet to other strutures. Referenes [] A. Mishenko n M. Perkowski, Logi synthesis of reversile wve ses, in Proeeings of IWLS, pp , [2] A. Khlopotine, M. Perkowski, n P. Kerntopf, Reversile logi synthesis y gte omposition, in Proeeings of IWLS, pp , [3] D. M. Miller, D. Mslov, n G. W. Duek, Synthesis of quntum multiple-vlue iruits, Journl of Multiple-lue Logi n Soft Computing, vol. 2, no. 5-6, pp , [4] G. Yng, W. Hung, X. Song, n M. Perkowski, Mjority-se reversile logi gtes, Theoretil Computer Siene, vol. 334, no. -3, [5] W. N. N. Hung, X. Song, G. Yng, J. Yng, n M. Perkowski, Optiml synthesis of multiple output oolen funtions using set of quntum gtes y symoli rehility nlysis, IEEE Trnstion on Computer-Aie Design of Integrte Ciruits n systems, vol. 25, no. 9, pp , 2006.

16 Synthesis of quntum iruits in Liner Nerest Neighor moel [6] N. Alhgi, M. Hwsh, n M. Perkowski, Synthesis of Reversile Ciruits for Lrge Reversile Funtions Ft Universitis, Series: Eletronis n Energetis, vol. 24, no. 3, pp , Deemer 200. [7] J. Fiurášek, Liner optil frekin gte se on prtil-swp gte, Phys. Rev. A, vol. 78, p , Sep [8] Y.-X. Gong, G.-C. Guo, n T. C. Rlph, Methos for liner optil quntum frekin gte, Phys. Rev. A, vol. 78, p , Jul [9] M. A. Nielsen n I. L. Chung, Quntum Computtion n Quntum Informtion. Cmrige University Press, [0] D. Chng, D. Mslov, n S. Severini, Trnsltion tehniques etween quntum iruits rhiteture. [] J. Cir n P. Zoller, Quntum omputtion with ol trppe ions, Physil Review letters, vol. 74, no. 20, p. 409, 995. [2] J. L. O Brien, G. J. Prye, A. G. White, T. C. Rlph, n D. Brnning, Demonstrtion of n ll-optil quntum ontrolle-not gte, Nture, vol. 426, pp , [3] D. E. Browne n T. Ruolph, Resoure-effiient liner optil quntum omputtion, Phys. Rev. Lett., vol. 95, p. 0050, Jun [4] A. Fowler, S. Devitt, n L. Hollenerg, Implementtion of shor s lgorithm on liner nerest neighor quit rry, Quntum Informtion n Computtion, vol. 4, no. 4, pp , [5] Y. Tkhshi, N. Kunihiro, n K. Oht, The quntum fourier trnsform on liner nerest neighor rhiteture, Quntum Informtion n Quntum Computtion, vol. 7, no. 4, pp , [6] S. Curo, T. Drper, S. Kutin, n D. Moutlon, A new quntum ripple-rry ition iruit. qunt-ph/04084, [7].. Shene, S. S. Bullok, n I. L. Mrkov, A prtil top-own pproh to quntum iruit synthesis, in Proeeings of Asi Pifi DAC, [8] A. Chkrrti n S. Sur-Koly, Nerest neighor se synthesis of quntum oolen iruits, Engineering Letters, vol. 5:2, p. 26, [9] A. Chkrrti n S. Sur-Koly, Rules for synthesizing quntum oolen iruits using minimize nerest-neighorhoo templtes, in Proeeings of the 5th Interntionl Conferene on Avne Computing n Communitions, pp , [20] Y. Hirt, M. Nknishi, n S. Ymshit, An effiient metho to onvert ritrry quntum iruits to ones on liner nerest neighor rhiteture, in 3 r Interntionl onferene on Quntum, Nno, n Miro Tehnologies, [2] M. Mozmmel H A Khn, N. Siik, n M. Perkowski, Minimiztion of Quternry Glois Fiel Sum of Prouts Expression for Multi-Output Quternry Logi Funtion using Quternry Glois Fiel Deision Digrm, in Proeeings of the Interntionl Symposium on Multiple-lue Logi 2008, [22] M. Mottonen n J. rtiinen, Deomposition of generl quntum gtes, in Trens in Quntum omputing Reserh, NOA, New York, [23] P. Muhri, Generl logi liner nerest neighor (llnn) rhiteture for fulttolernt quntum omputtion, Mster s thesis, Wihit Stte University, [24] R. Wille, M. Seei, n R. Drehsler, Synthesis of reversile funtions eyon gte ount n quntum ost, in Proeeings of the IWLS, 2009.

17 86 M. Perkowski, M. Luk, D. Shh, n M. Kmeym: [25] D. Wineln n T. Heinrihs, Ion trp pprohes to quntum informtion proessing n quntum omputing, A Quntum Informtion Siene n Tehnology Romp, vol. N/A, [26] D. Mslov n G. Duek, Improve quntum ost for n-it Toffoli gtes, IEE Eletroni Letters, vol. 39, no. 25, pp , [27] D. Mslov n G. Duek, Grge in reversile esign of multiple output funtions, in proeeings of the 6th Interntionl Symposium on Representtions n Methoology of Future Computing Tehnologies, pp , [28] M. Luk, M. Perkowski, H. Goi, M. Pivtoriko, C. H. Yu, K. Chung, H. Jee, B.-G. Kim, n Y.-D. Kim, Evolutionry pproh to quntum reversile iruit synthesis, Artif. Intell. Review., vol. 20(3-4), pp , [29] D. Mslov, G. W. Duek, n D. M. Miller, Synthesis of Frekin-Toffoli reversile networks, IEEE Trnstions on LSI, vol. 3, no. 6, pp , [30] A. Agrwl n N. Jh, Synthesis of reversile logi, in Proeeings of DATE, pp , [3] R. Wille n R. Drehsler, B-se synthesis of reversile logi for lrge funtions, in Proeeings of the 46th Annul Design Conferene, [32] M. Luk, M. Perkowski, n M. Kmeym, Evolutionry quntum logi synthesis of oolen reversile logi iruits emee in ternry quntum spe using struturl restritions, in Proeeings of the WCCI 200, 200. [33] A. Breno, C. H. Bennett, R. Cleve, D. P. Diinenzo, N. Mrgolus, P. Shor, T. Sletor, J. A. Smolin, n W. H., Elementry gtes for quntum omputtion, Physil Review A, vol. 52, pp , 995. [34] M. Perkowski, L. Jozwik, n R. Drehsler, A nonil n/exor form tht inlues oth the generlize ree-muller forms n kroneker ree-muller forms, in Pro. of the Ree-Muller 997 Conferene, pp , 997. [35] m. Chrznowsk-Jeske, Z. Wng, n Y. Xu, Regulr representtion for mpping to fine-grin, lolly-onnete fpgs, in Proeeings of ISCAS, 997. [36] D. Shh n M. Perkowski, Synthesis of quntum rrys with low quntum osts from kroneker funtionl lttie igrms, in IEEE Congress on Evolutionry Computtion, pp. 7, 200. [37] M. Perkowski, M. Chrznowsk-Jeske, n Y. Xu, Lttie igrms using reemuller logi, in Proeeings of RM, 997. [38] M. Perkowski, A. Al-Ri, P. Kerntopf, A. Buller, M. Chrznowsk-Jeske, A. Mishhenko, M. M. Az Khn, A. Coppol, S. Ynushkevih,. Shmerko, n L. Jozwik, A generl eomposition for reversile logi, in Pro. RM 200, August 200. [39] M. Perkowski, P. Kerntopf, A. Buller, M. Chrznowsk-Jeske, A. Mishenko, X. Song, A. Al-Ri, L. Jozwik, A. Coppol, n B. Mssey, Regulr reliztion of symmetri funtions using reversile logi, in Proeeings of EUROMICRO Symposium on Digitl System, 200. [40] S. Akers, A retngulr logi rry, IEEE Trnstions on Computers, vol. C-2, pp , 972. [4] M. A. Perkowski, M. Chrznowsk-Jeske, n Y. Xu, Lttie igrms using reemuller logi, in IFIP WG 0.5 Workshop on Applitions of the Ree-Muller Expnsion in Ciruit Design, pp , 997. [42] M. Chrznowsk-Jeske, Z. Wng, n Y. Xu, A regulr representtion for mpping to fine-grin, lolly-onnete fpgs, in Ciruits n Systems, 997. ISCAS 97.,

18 Synthesis of quntum iruits in Liner Nerest Neighor moel Proeeings of 997 IEEE Interntionl Symposium on, vol. 4, pp vol.4, June 997. [43] S. Lee, S. Kim, J. Bimonte, n M. Perkowski, The ost of quntum gte primitives, Journl of Multiple-lue Logi n Soft Computing, vol. 2, no. 5-6, pp , [44] M. Luk, rile orering using g for lttie igrms n y gtes iruits. tehnil report, 200.