Reducing the Depth of Quantum Circuits Using Additional Circuit Lines

Transcription

1 Ruing th Dpth of Quntum Ciruits Using Aitionl Ciruit Lins Nil Assi 1, Rort Will 1,2, Mthis Sokn 1,2, n Rolf Drhslr 1,2 1 Institut of Computr Sin, Univrsity of Brmn Group of Computr Arhittur, D Brmn, Grmny 2 Cyr-Physil Systms, DFKI GmH D Brmn, Grmny {nil,rwill,msokn,rhsl}@informtik.uni-rmn. Astrt. Th synthsis of Booln funtions, s thy r foun in mny quntum lgorithms, is usully onut in two stps. First, th funtion is rliz in trms of rvrsil iruit follow y mpping into orrsponing quntum rliztion. During this pross, th numr of lins n th quntum osts of th rsulting iruits hv minly n onsir s optimiztion ojtivs thus fr. Howvr, yon tht lso th pth of quntum iruit is vitl. Although first synthsis pprohs tht onsir pth hv rntly n introu, th mjority of sign mthos i not onsir this mtri. In this ppr, w introu n optimiztion pproh iming for th rution of pth in th pross of mpping rvrsil iruit into quntum iruit. For this purpos, w prsnt n improv (lol) mpping of singl gts s wll s (glol) optimiztion shm onsiring th whol iruit. In oth ss, w inorport th i of xploiting itionl iruit lins whih r us in orr to split hin of sril gts. Our optimiztion thniqus nl onurrnt pplition of gts whih signifintly rus th pth of th iruit. Exprimnts show tht rutions of pprox. 40% on vrg n hiv whn following this shm. 1 Introution Quntum omputtion hs om n tiv rsrh fil u to its promising rsults for importnt tsks suh s ftoriztion or ts srh. Motivt y this, rsrhrs hv vlop svrl synthsis pprohs [1 5]. Mny quntum lgorithms r oftn sri y mns of strutur quntum iruit in whih only th rprsnttion of Booln omponnts iffrs. Hn, for th synthsis of ths omponnts into quntum iruits, usully two-stp pproh is ppli: First, th sir Booln funtionlity is rliz s rvrsil iruit only onsisting of rvrsil gts whih is ftrwrs mpp to n quivlnt rliztion s on quntum gts. For this purpos, mpping shms s introu.g. in [6, 15] r ppli. In this flow, minimizing th numr of lins n th quntum osts hv n onsir s th mjor optimiztion ojtivs thus fr. Howvr, yon tht

2 lso th pth of th iruit is vitl. Dpth optimiztion thniqus onsir th onurrnt pplition of singl gts in orr to ru th ovrll xution tim of th iruit rliztion. Whil first pprohs for synthsis with rspt to pth hv rntly n introu (s.g. [7 11]), th vst mjority of sign mthos os not onsir this mtri. As n xmpl in [7, 11], yl rprsnttion ws hosn n input yls whr prtition into thr susts. Eh sust is synthsiz inpnntly on iffrnt st of nill in prlll. This mtho rquirs 2n itionl lins n fouss only on ruing th pth of rvrsil iruit rthr thn th quntum iruit. This is ruil sin th xution tims for two rvrsil gts n iffr signifintly whn tking th rsptiv quntum iruit mpping into ount. As onsqun, vn pth-optiml rvrsil iruit likly ls to quntum iruit with non-optiml pth. Anothr postsynthsis pproh hs n prsnt in [8]. Howvr, thir pproh mks us of spil lss of tmplts. Finlly, th work prsnt in [10] sris n xhustiv lgorithm iming to fin miniml pth quntum iruit using spil gt lirry. Howvr, u to its xponntil tim omplxity, it is only pplil to iruits with smll numr of quits. In this ppr, w prsnt n i on how pth of quntum iruits n ru y ing n itionl lin to th iruit. Bs on this i, two pth optimiztion pprohs r prsnt. Th first mtho ims to ru th pth y pplying th rution gt-pr-gt, whrs th son mtho fouss on th whol iruit. An xprimntl vlution of oth pprohs shows tht signifint improvmnt of pth n hiv for quntum iruits. Th rminr of this ppr is strutur s follows. Th nxt stion rifly introus rvrsil n quntum iruits. Dpth mtris n th gnrl i r prsnt in St. 3. Aftrwrs, oth propos pprohs r sri n vlut in St. 4 n St. 5, rsptivly. Finlly, St. 6 onlus th ppr. 2 Bkgroun To kp th rminr of this ppr slf-ontin, this stion rifly introus th sis on rvrsil iruits, quntum iruits, n th orrsponing mpping from rvrsil to quntum iruits. 2.1 Rvrsil Ciruits Booln rvrsil funtions r thos funtions f : IB n IB n tht r ijtiv, i.. thr xists n 1-to-1 mpping from th inputs to th outputs n vi vrs. Rvrsil funtions n rliz y rvrsil iruits tht onsist of t lst n lins. Rvrsil iruits r ss of rvrsil gts tht long to gt lirry. On gt lirry tht is oftn us onsists of multipl ontrol Toffoli gts [12].

3 () Rvrsil gt () Rvrsil iruit Fig. 1. Rvrsil iruitry Dfinition 1. Givn st of vrils V = {x 1,..., x n }, multipl ontrol Toffoli gt T(C, t) hs ontrol lins C = {x j1, x j2,..., x jl } V n trgt lin t V \ C. Th gt mps t t (x j1 x j2 x jl ) n lvs ll othr lins unltr. In th spil ss C = 0 n C = {} = 1, th gts r rfrr to s NOT n CNOT gt n not N(t) n C(, t), rsptivly. In [13], it hs n shown tht ny rvrsil funtion f : IB n IB n n rliz y rvrsil iruit with n lins whn using Toffoli gts. Exmpl 1. Figur 1() shows Toffoli gt with two ontrol lins. Th ontrol lins r not y, whil th trgt lin is not y. Th nnott vlus monstrt th omputtion of th gt for givn input ssignmnt. Figur 1() shows iffrnt Toffoli gts in s forming rvrsil iruit. 2.2 Quntum Booln Ciruits Inst of its, quntum iruits mnipult quits whih n rprsnt th lssil Booln vlus ut lso th suprposition of thm. Mor prisly, quit ϕ is vtor ( ) whr, C suh tht = 1. If = 1, thn ϕ rprsnts th lssil 0, not 0, n if = 1, thn ϕ rprsnts th lssil 1, not 1. In gnrl, quntum gt ting on n quits rprsnts 2 n 2 n unitry mtrix [14], whr mtrix U is unitry if U U = UU = I n U is th joint mtrix U = U T. Using this gt finition, mny quntum mhnil ffts suh s suprposition n ntnglmnt n formult. Howvr, in th sop of this ppr w r onsiring iruits tht rliz pur Booln funtionlity ut still n to rliz using quntum gts in orr to m thm into quntum lgorithms suh s Dutsh-Josz, Grovr, or Shor. Toffoli gts rprsnt unitry mtrix n r hn suitl for rlizing quntum Booln iruits. Howvr, with rspt to th tul physil implmnttion, it is of intrst to otin iruits tht mk us of gts from lirry with only fw lmnts [6]. For th prsnt ppr, w r mking us of ommon gt lirry onsisting of four quntum gts tht only hng on quit t tim n is fin s follows. Dfinition 2. A quntum gt U(C, t) pplis th unitry 2 2 mtrix to th quit tht orrspons to th trgt lin t, if n only if ll ontrol lins C r s-

4 V 1 V v 0 V 0 Fig. 2. Quntum iruitry x 1 x 1 x 2 x 2 x 3 V V V x 3 x 1x 2 () Bs on Fig. 1() x 1 x 1 x 2 x 2 x 3 x 3 x 4 V V V V V V x 4 () Bs on Fig. 1() Fig. 3. Mpping rvrsil iruits to quntum iruits sign 1. W ( onsir ) gt lirry X({}, t), X({}, t), V ({}, t), n V ({}, t) 0 1 with X =, V V = X, n V 1 0 ing th joint of V. Not tht X({}, t) = N(t) n X({}, t) = C(, t). Th gt lirry is oftn rfrr to s NCV lirry. Exmpl 2. Figur 2 pits quntum iruit onsisting of four gts, whr v 0 = V 1 = V Mpping Rvrsil Ciruits to Quntum Ciruits Sin ny quntum oprtion n rprsnt y unitry mtrix [14], h quntum iruit is inhrntly rvrsil. As isuss ov, whn rvrsil iruits shoul rprsnt s quntum iruit, th Toffoli gts r too gnrl n, thus, not suitl for rliztion. As onsqun, rvrsil iruits r mpp to quntum iruits tht only onsists of gts of prtiulr gt lirry,.g. th NCV lirry. For this purpos, h gt of th rvrsil iruit is mpp into s of funtionlly quivlnt quntum gts. Exmpl 3. Consir Toffoli gt with two ontrol lins s shown in Fig. 1(). A funtionlly quivlnt rliztion in trms of quntum gts is pit in Fig. 3(). This s n ppli to fully mp th rvrsil iruit shown in Fig. 1() into n quivlnt quntum iruit. For this purpos, ll orrsponing Toffoli gts r rsptivly sustitut with orrsponing quntum gt s. Th 2 n, 4 th, n 5 th gt rmin unhng s thy lry rprsnt quntum gts. Th rsulting fully quivlnt quntum iruit is shown in Fig. 3(). Similr mppings xist for Toffoli gts with mor thn two ontrol lins. But with inrsing numr of ontrol lins, th rsulting quntum iruits om

5 mor xpnsiv, i.. rquir mor quntum gts. Th urrntly st known mppings of singl Toffoli gts into quntum ss hv n introu in [15]. In this work, w r following th mppings introu thr. As singl quntum gts r ssum to hv unit osts, th numr of gts of th rsulting ss usully is rfrr to s quntum osts. 3 Ruing th Dpth of Quntum Ciruits In this work, w r proposing optimiztion pprohs iming for rution of th pth in quntum iruits using itionl iruit lins. This stion first motivts th onsirtion of pth in quntum iruits, whrs th gnrl i of th propos pprohs is outlin ftrwrs. 3.1 Consirtion of Dpth in Quntum Ciruits Thus fr, th mjor optimiztion ojtivs for synthsis hv n th numr of lins n th quntum osts of th rsulting iruits s rviw ov. Howvr, yon tht lso th pth of quntum iruit is vitl. This mtri rognizs whthr gts n onurrntly ppli whih likly ls to rution in th xution tim of iruit. Dfinition 3. Lt U i (C i, t i ) n U i+1 (C i+1, t j+1 ) two onsutiv quntum gts. Ths gts n ppli onurrntly if C i C i+1 {t i, t i+1 } = C i + C i In othr wors, if th lins us y h gt (oth ontrol n trgt lin) r isjoint. Lt G quntum iruit with k lmntry quntum gts, thn G n prtition into m k suiruits whos gts n pirwis ppli onurrntly. W rfr to th miniml m s th pth of th iruit. Algorithm D (Dtrmin Ciruit Dpth). Givn quntum iruit G = U 1 (C 1, t 1 )... U k (C k, t k ) ovr n vrils x 1,..., x n. This lgorithm trmins th pth m of th iruit oring to Dfinition 3 y pplying gry srh to gts tht n xut in prlll. For th omputtion, w r mking us of th intgrs 1,..., n. D1. [Initiliz.] St m 1, i 1, n j 0 for 1 j n. D2. [Trmint?] If i > k, trmint. D3. [Apply gt.] For h x j C i {t i }, st j j + 1. D4. [Gts o not ovrlp?] If thr xists no j {1,..., n} suh tht j = 2, st i i + 1 n goto Stp D2. D5. [Gts ovrlp.] For h j {1,..., n}, st j 1, if x j C i {t i }, othrwis st j 0; st m m + 1, i i + 1, n goto Stp D2. Exmpl 4. Figur 4 illustrts th pth for th rvrsil iruit shown in Fig. 1().

6 V V V V V V Fig. 4. Quntum pth for th rvrsil iruit shown in Fig. 1() Although th ohrn tim, i.. th tim quit n kp its quntum stt, n th gt oprtion tim, i.. th tim gt ns to prform its oprtion, my vry from on thnology to nothr (s.g. Tl III of [16]), kping th ovrll xution tim s smll s possil is ssntil in ll ths ss. Consquntly, th pth mtri n ppli in gnri mnnr, s it provis propr mol whih n onsir lry t th synthsis stg in th sn of pris thnologil onstrints. Dspit th ft tht quntum lgorithms lry xploit lgorithmi prlllism to inrs th prossing sp, synthsis pprohs shoul im for prouing iruits with t lst s possil iruit pth. Motivt y this, w r onsiring th qustion how th pth of quntum iruit n ru in th rminr of this ppr. For this purpos, w r mking us of itionl iruit lins s motivt in th following. 3.2 Exploiting Aitionl Ciruit Lins Kping th numr of iruit lins s smll s possil is wll pt in th synthsis of quntum iruits. This is minly motivt y th ft tht h iruit lin hs to rprsnt y quit, whih is vry limit rsour. Nvrthlss, vlutions lso show tht (slight) xtnsion of iruit with itionl lins my hv signifint nfits. For xmpl in [6, 15], it hs n monstrt tht lrgr mount of iruit lins llow for muh hpr mpping of rvrsil iruits to quntum iruits in trms of gt ount. In [3], vlutions show tht using twi th numr of iruit lins rus th quntum osts y up to two orrs of mgnitu. Evntully, this l to post-synthsis optimiztion pproh [17] whih nls rutions in quntum osts of up to 69% only y ing singl itionl lin to th iruit. In this work, w show tht similr onpts lso hlp in ruing th pth of quntum iruits. W r following th stlish synthsis flow rviw in St. 2.3, i.. first rvrsil iruit is rliz whih ftrwrs is mpp to quntum iruit. Howvr, y inorporting itionl lins uring this pross, pth-wr optimiztion oms possil. Th itionl iruit lins r introu s hlpr lins. Dfinition 4. Lt G rvrsil or quntum iruit. A hlpr lin is n itionl lin whos input is st to onstnt 0 n is us in wy throughout th iruit suh tht th output of th lin is lso 0.

7 V V V V V V V V V V V V 0 0 () Initil iruit () Ciruit with ru pth Fig. 5. Dpth rution y using itionl hlpr lins Following th onpt from [17], hlpr lins n now ppli in orr to uffr vlus of iruit lins so tht thy n r-us ltr y othr gts. Whnvr th urrnt vlu of hlpr lin h is 0, nothr signl lin x n opi to h y ppning opy gt C({x}, h) to th iruit. Th hlpr lin n rstor with th sm gt if no othr gt hs us h s trgt lin in twn. In [17], this uffring hs n xploit to rmov ommon ontrol lins onntions twn Toffoli gts in orr to ru th quntum ost. Howvr, th sm onpt n similrly ppli to ru th pth of quntum iruits s illustrt y th following xmpl. Exmpl 5. Figur 5() shows iruit in whih no gts n prform in prlll sin thy ll shr th sm ontrol lin. In Fig. 5() hlpr lin hs n to opy th vlu of. By oing this, th gts n rrrng whih rus th pth from 8 to 6. Clrly, Exmpl 5 prsnts rthr rtifiil iruit. Howvr, s on this gnrl i w r proposing iffrnt optimiztion pprohs whos vlutions show tht in signifint rution of pth in quntum iruits n hiv. 4 Optimiztion Approhs Motivt y th gnrl i outlin ov, two optimiztion pprohs r propos in this stion whih im for ruing th pth y xploiting itionl iruit lins. Th first pproh follows lol shm, i.. onsirs h Toffoli gt inpnntly, whr th son pproh onsirs th whol iruit inst. Finlly, thniqus r prsnt to furthr ru th pth n th quntum osts whih n ppli to th rsulting quntum ss. 4.1 Consirtion of Singl Toffoli Gts Th vilility of hlpr lin s introu in th prvious stion llows for n improvmnt of th mpping shm rviw in St Rll tht, whn

8 () Toffoli Gt V V V () Originl mpping V V V 0 0 () Propos mpping Fig. 6. Consirtion of singl Toffoli gts following th fult mpping shm, h Toffoli gt is mpp to quntum rliztion of pth 5 s shown in Fig. 6(). Howvr, s th son n th thir gt shr th sm ontrol lin, n itionl hlpr lin llows for onurrnt xution of oth gts s shown in Fig. 6(). Sin itionlly th opy gts n insrt without inrsing th pth, pth rution for th quntum iruit rliztion for h Toffoli gt from 5 to 4 n otin. Exmpl 6. Consir gin th rvrsil iruit from Fig. 1(). Using th stlish mpping shm from St. 2.3, quntum iruit with pth 12 rsults (s shown in Fig. 3(); non of th gts xpt for th singl NOT gt n xut onurrntly). In ontrst, pplying th itionl hlpr lin s propos in Fig. 6, th iruit pit in Fig. 7() rsults. This rus th pth from 12 to 9. Not tht this prour n lso ppli to Toffoli gts with mor thn two ontrol lins. In ft, stt-of-th-rt mpping shms (suh s sri in [15]) ompos ths gts into ss of two-ontroll Toffoli gts. For thm, th pth-optimiz mpping to quntum gts s propos in Fig. 6 n ppli. Morovr, th sm shm n ppli to othr rvrsil gts suh s th Prs gt s wll. This shm is not nfiil in ll ss. In ft, if onurrnt Toffoli gts r mpp to quntum iruit, th originl mpping ls to ttr rsults. This is illustrt y mns of Fig. 8. Applying th originl mpping shm to th two Toffoli gts shown in Fig. 8() ls to th quntum s s shown in Fig. 8(). As oth Toffoli gts r ppli onurrntly, lso th rsulting quntum gt ss n ppli onurrntly, i.. pth of 5 rsults. Applying th propos shm from Fig. 6 woul worsn th rsult. In ft, th 0 V V V V V V 0 () Consirtion of singl Toffoli gts 0 V V V V V V 0 () Consirtion of th whol iruit Fig. 7. Applition of th propos pprohs to th iruit from Fig. 1()

9 f f f V V V V V V f V V V f V V V f 0 0 () Originl iruit () Originl mpping () Propos mpping Fig. 8. Applition of th lol shm to onurrnt Toffoli gts V V V () Originl iruit V V V 0 0 () Rsulting iruit Fig. 9. Consirtion of th whol iruit hlpr lin togthr with th rquir opy gts woul inrs th pth to 7 s shown in Fig. 8(). Consquntly, this shm is only ppli in ss whr n tul pth improvmnt n hiv. Howvr, xprimnts summriz in St. 5 lrly onfirm tht sustntil improvmnts with rspt to th pth n still hiv. As rwk, this oviously oms with th pri of inrs quntum osts in th rsulting s. But lso hr, xprimnts show th rsulting inrs to mort. 4.2 Consirtion of th Whol Ciruit Whil so fr th hlpr lin hs n xploit in lol ontxt, lso glol onsirtion turns out to nfiil. Th i is to intify suiruits of gts shring th sm ontrol lin n us th hlpr lin in orr to prtition th gts. Thn, h onsutiv pir of gts in suh s n onurrntly xut y using th originl ontrol lin for th first gt n th opi vlu t th hlpr lin for th son gt. Exmpl 7. Figur 9() shows quntum iruit ompos of gts tht shr th sm ontrol lins. Using th hlpr lin, n quivlnt rliztion s shown in Fig. 9() n riv. This rus th pth from 5 to 4. This shm n itionlly improv y pplying th moving rul for quntum iruits. In ft, two jnt gts U(C 1, t 1 ) n U(C 2, t 2 ) n intrhng if t 2 / C 1 n t 1 / C 2 {t 1 } =. As rsult, gts n mov through th iruit whih might l to lrgr suiruits of gts shring th sm ontrol lin. In this s, mor sustntil rution n hiv.

10 Exmpl 8. Consir gin th quntum iruit shown in Fig. 4. Th son, fifth, sixth, n svnth gt shr th sm ontrol lin n n mov togthr (not, lthough lso th thir n tnth gt hv ontrol lin, thy nnot mov to onsutiv s). Exploiting tht, this s n optimiz ling to th iruit shown in Fig. 7(). This rus th pth from 12 to 9. Not tht this shm lso inrss th quntum osts of th rsulting iruit. Howvr, sin for h intifi suiruit only two opy gts hv to, th inrs is lmost ngligil. 4.3 Furthr Optimiztions Inpnnt of th optimiztion shms propos ov, th pth of quntum gt ss n itionlly improv using xisting optimiztion shms tht originlly im for quntum ost rution. In prtiulr, th pplition of mrging n ltion ruls s xplin in [18] togthr with th moving rul s lry isuss ov is nfiil. For xmpl, th iruit shown in Fig. 7() (otin using xisting mpping shms) oviously n improv y rmoving th fifth n th sixth gt whih nl h othr. This rus th quntum osts ut lso improvs th pth of th iruit. Aoringly, suh simpl optimiztions r lso ppli in our pproh. For th xprimntl vlution summriz in St. 5, th mthos xploiting itionl hlpr lins r ppli to iruits lry optimiz using moving, mrging, n ltion rul. 5 Exprimntl Rsults In orr to onfirm th ffiiny of th propos i, th pprohs sri ov hv n implmnt n xprimntlly vlut. For this purpos, th opn sour toolkit RvKit [19] hs n ppli n nhmrks hv n tkn from th RvLi [20] ts. All xprimnts hv n onut on n Intl Cor i5 Prossor with 4 GB of min mmory. In this stion, w summriz n isuss th otin rsults. Tl 1 provis th otin numrs. For ll nhmrks list in th first olumn, th numr of lins (Lins), th quntum osts (Costs), n th pth (Dpth) of th rsptiv iruit rliztions s wll s th run-tim (Tim) n to gnrt thm r provi. W istinguish twn th following iruits: Initil Ciruits (IC) rprsnt th iruits s tkn from RvLi n mpp to quntum iruits s sri in St. 2.3, i.. without ny pth optimiztion whtsovr. Optimiz Ciruits (OC) rprsnt th iruits tht hv itionlly n optimiz using th strightforwr thniqus rviw in St. 4.3.

11 Both, th initil iruits n optimiz iruits, llow for omprison to th iruits otin y th propos thniqus, nmly: Ciruits tht hv n otin y using th optimiztion shm tht onsirs singl Toffoli gts (Lol) s sri in St Ciruits tht hv n otin y using th optimiztion shm tht onsirs th whol iruit (Glol) s sri in St Th prntg pth-improvmnt of th iruits otin y th propos thniqus with rspt to th initil iruit n th optimiz iruits r provi in th olumns not y Impr IC n Impr OC, rsptivly. First of ll, it n osrv tht lry th niv pprohs rviw in St. 4.3 l to signifint improvmnts (15% on vrg n up to 42% in th st s for 4mo5-v0 18 ). Howvr, xploiting itionl iruit lins nls furthr improvmnts whih r ftors yon tht. In th st s (x5p 296 ), pth n ru from 1352 to 303 (using th lol pproh from St. 4.1) or 226 (using th glol pproh from St. 4.2). But lso for th othr nhmrks sustntil rutions n osrv, vn ompr to th lry optimiz iruits. As isuss ov, ths improvmnts in th pth my om t th pri of highr quntum osts. As our vlutions show, this prtiulrly hols for th lol onsirtion of singl Toffoli gts (s olumns not Lol). Hr, quntum osts inrs y 18% on vrg ompr to th lry optimiz iruit. Howvr, for th glol shm, no suh isvntgs n osrv. In ft, quntum osts rmin unhng hr (s olumns not Glol). Ovrll, vn ompr to lry optimiz iruits, improvmnts of mor thn 50% on vrg n hiv. If th glol shm is ppli, ths hivmnts r possil without th n to pt n inrs in th quntum osts. This is m possil y th ition of singl iruit lin. Although this vntully rsults in th onsirtion of nothr quit to physilly rliz, th possil nfits with rspt to timing n prtiulrly ohrn tim might worth th ovrh. 6 Conlusion In this ppr, pth optimiztion y ing hlpr lin to quntum iruits hs n introu n vlut. Two pprohs, nmly gt s n iruit s, hv n onsir. Exprimntl rsults for th two mthos hv shown signifint pth rutions whih rhs ovr 50% for quntum iruits. Although ths mthos inrs quntum ost, pplying furthr improvmnts to th quntum iruits hv fix th prolm.

12 Tl 1. Exprimntl vlution Bnhmrk Initil Ciruits Optimiz Ciruits Lol Glol (St. 4.3, +0 lin) (St. 4.1, +1 lin) (St. 4.2, +1 lin) Lins Cost Dpth Tim Cost Dpth Tim ImprIC Cost Dpth Tim ImprIC ImprOC Cost Dpth Tim ImprIC ImprOC x5p , ,95 15% ,48 81% 78% ,92 86% 83% hw , ,71 15% ,64 78% 74% ,58 84% 81% , ,04 6% ,73 75% 73% ,66 75% 73% hw , ,88 14% ,59 76% 72% ,21 83% 80% hw , ,09 14% ,28 71% 66% ,44 78% 75% w , ,26 14% ,11 71% 66% ,47 76% 72% hw , ,67 12% ,79 68% 64% ,42 74% 71% , ,23 20% ,06 69% 62% ,99 69% 62% , ,06 19% ,35 69% 61% ,85 68% 61% , ,02 19% ,19 67% 60% ,14 67% 59% hm , ,51 25% ,35 70% 60% ,22 72% 63% , ,00 17% ,03 65% 57% ,03 63% 56% hw , ,11 12% ,15 61% 56% ,95 65% 60% mo5r , ,17 13% ,80 62% 56% ,55 69% 65% r , ,08 15% ,80 62% 55% ,73 66% 60% r , ,30 16% ,65 62% 55% ,79 72% 67% , ,00 17% ,58 57% 48% ,99 57% 48% nt , ,00 5% ,03 50% 47% ,04 39% 36% hm , ,03 23% ,14 55% 42% ,12 60% 49% mo , ,00 7% ,00 29% 23% ,00 36% 31% urf , ,19 5% ,74 26% 22% ,03 28% 24% 4mo5-v , ,00 42% ,00 54% 21% ,00 63% 36% ryy , ,36 12% ,14 28% 18% ,55 29% 19% lu , ,19 12% ,46 27% 18% ,25 29% 19% Avrg Improvmnt 15% 60% 52% 63% 56% Lgn of th tl is sri in Stion 5.

13 Rfrns 1. Fzl, K., Thornton, M., Ri, J.: ESOP-s toffoli gt s gnrtion. In: Communitions, Computrs n Signl Prossing, PRim Pifi Rim Confrn on. (2007) Millr, D.M., Mslov, D., Duk, G.W.: A trnsformtion s lgorithm for rvrsil logi synthsis. In: Dsign Automtion Conf. (2003) Will, R., Drhslr, R.: BDD-s synthsis of rvrsil logi for lrg funtions. In: Dsign Automtion Conf. (2009) Sokn, M., Will, R., Ottrstt, C., Drhslr, R.: A synthsis flow for squntil rvrsil iruits. In: Int l Symposium on. Multipl-Vlu Logi (2012) Sokn, M., Will, R., Hilkn, C., Przigo, N., Drhslr, R.: Synthsis of rvrsil iruits with miniml lins for lrg funtions. In: ASP Dsign Automtion Conf. (2012) Brno, A., Bnntt, C.H., Clv, R., DiVinhnzo, D., Mrgolus, N., Shor, P., Sltor, T., Smolin, J., Winfurtr, H.: Elmntry gts for quntum omputtion. Th Amrin Physil Soity 52 (1995) Arzh, M., Sh Zmni, M., Sighi, M., Si, M.: Dpth-optimiz rvrsil iruit synthsis. Quntum Informtion Prossing (2012) Mslov, D., Duk, G., Millr, D., Ngrvrgn, C.: Quntum iruit simplifition n lvl omption. Trnstions on Computr-Ai Dsign of Intgrt Ciruits n Systms 27(3) (2008) Bohrov, A., Svor, K.M.: A pth-optiml nonil form for singl-quit quntum iruits. rxiv prprint rxiv: (2012) 10. Amy, M., Mslov, D., Mos, M., Rottlr, M.: A mt-in-th-mil lgorithm for fst synthsis of pth-optiml quntum iruits. rxiv prprint rxiv: (2012) 11. Arzh, M., Zmni, M., Sighi, M., Si, M.: Logil-pth-orint rvrsil logi synthsis. In: Int l Workshop on Logi n Synthsis. (2011) 12. Toffoli, T.: Rvrsil omputing. In Bkkr, W., vn Luwn, J., s.: Automt, Lngugs n Progrmming. Springr (1980) 632 Thnil Mmo MIT/LCS/TM-151, MIT L. for Comput. Si. 13. Shn, V.V., Prs, A.K., Mrkov, I.L., Hys, J.P.: Synthsis of rvrsil logi iruits. Trnstions on Computr-Ai Dsign of Intgrt Ciruits n Systms 22(6) (2003) Nilsn, M., Chung, I.: Quntum Computtion n Quntum Informtion. Cmrig Univ. Prss (2000) 15. Millr, D.M., Will, R., Ssnin, Z.: Elmntry quntum gt rliztions for multipl-ontrol Toffolli gts. In: Int l Symp. on Multi-Vlu Logi. (My 2011) Mtr, R.V., Oskin, M.: Arhitturl implitions of quntum omputing thnologis. J. Emrg. Thnol. Comput. Syst. 2(1) (2006) Millr, D.M., Will, R., Drhslr, R.: Ruing rvrsil iruit ost y ing lins. In: Int l Symp. on Multi-Vlu Logi. (2010) Zhr, S.: Thnology Mpping n Optimiztion for Rvrsil n Quntum. PhD thsis, Univrsity of Vitori (2012) 19. Sokn, M., Frhs, S., Will, R., Drhslr, R.: Rvkit: n opn sour toolkit for th sign of rvrsil iruits. Rvrsil Computtion, Ltur Nots in Computr Sin 7165 (2012) RvKit is vill t

14 20. Will, R., Groß, D., Tur, L., Duk, G.W., Drhslr, R.: RvLi: n onlin rsour for rvrsil funtions n rvrsil iruits. In: Int l Symp. on Multi- Vlu Logi. (2008) RvLi is vill t