Nearest-Neighbor and Fault-Tolerant Quantum Circuit Implementation

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1 Neres-Neighbor nd Ful-olern Qunum Circui Implemenion Lxmidhr Biswl, Chndn Bndyopdhyy, Anupm Chopdhyy 2, Rober Wille 3, Rolf Drechsler 4, fizur Rhmn Indin Insiue of Engineering cience nd echnology hibpur, Indi Nnyng echnologicl Universiy, ingpore, 50 Nnyng Ave, ingpore Insiue for Inegred Circuis, Johnnes Kepler Universiy Linz, A-4040 Linz, Ausri 4 Insiue of Compuer cience, Universiy of Bremen & Cyber-Physicl ysems, DFKI Gmb, Bremen, Germny Emil: lxmidhr.cvrce@gmil.com, chndnb@i.iiess.c.in, nupm@nu.edu.sg, rober.wille@jku., drechsle@uni-bremen.de, rhmn_h@i.iiess.c.in Absrc he ques of chieving higher compuing performnce is driving he reserch on qunum compuing, which is reporing new milesones lmos on dily bsis. For prcicl qunum circui design, ful olernce is n essenil condiion. his is chieved by mpping he rge funcions ino he Clifford group of elemenry qunum ges. Furhermore, he pplicion of error-correcing codes in qunum circuis requires he qunum ges o be formed beween djcen Qubis. In his work, we improve he se-of-he-r qunum circui design by ddressing boh of he bove chllenges. Firs, we propose novel mpping of Muliple-Conrol offoli (MC) ges o Clifford group ges, which chieves lower ge coun compred o erlier work. econdly, we show generic wy o conver ny Clifford circui ino neres neighbor one. We vlide he efficcy of our pproch wih deiled experimenl sudies. Keywords Qunum Cos, MC (Muliple Conrol offoli), Clifford groups, NNC (Neres Neighbor Cos) I. INRODUCION Qunum compuing [-2] is one of he mos promising compuing echnologies wih esblished heoreicl resuls proving h here exiss significn gp beween he clssicl nd qunum compuing [3]. In erms of prcicl relizion, new cpbiliies in devices [4] nd circuis [5] re being repored on regulr bsis. Qunum circuis re relized by severl echnologies such s, Ion rp [6], Nucler Mgneic Resonnce (NMR) [7], Kne model [8] nd superconducing circuis [9]. In recen brekhrough, 2-qubi qunum ges hve been relized in ilicon [0]. One of he mjor hindrnces owrds prcicl qunum circui relizion is ful-olernce []. his is ddressed he logic level wih ful deecion sregies [2]. A he circui level, error correcing codes re combined wih physicl qubis o provide mximum fideliy. Consequenly, he universl operor se of Clifford [3] is chosen s he mos suible se due o he known consrucions of Clifford group of operors nd ge for mos promising error correcing codes, including surfce codes. For surfce codes, i is lso necessry h he enngled ges re rrnged in neres neighbor fshion. herefore, n efficien mpping of given funcion o Clifford circui wih neres neighbor consrucion is n imporn design chllenge. A. Preliminries Definiion : he qunum se of qubi ψ is he superposiion of he wo bsic ses 0 nd which is defined s ψ α 0 β, where α nd β re he complex numbers h define probbiliy vlues for ses 0 nd, respecively, such h α 2 β 2. Ech of he qunum ges, which re represened by uniry mrices, c on qubis. For k-qubi qunum ge, he size of he uniry mrix is 2 k x 2 k. he number of qubis in he inpu nd oupu of he ge re he sme nd ll qunum ges re funcionlly reversible. he funcionliy of ech qunum ge is obined by muliplying he uniry rnsformion mrix wih he qunum se vecor. In ble, we show some of he common single-qubi nd wo-qubi qunum ges ogeher wih heir operionl behvior. Definiion 2: A qunum ge librry is collecion of qunum ges like NO, CNO, V/V, W/W ec. h performs he logicl mpping beween qunum ges o universl reversible ges (MC ges). NCV, NCVW, NCV v > re well known ge librries [4-7] h cn relize MC ges. For exmple, qunum ge relizion of he wo conrol offoli ge using he NCV ge librry is shown in Fig. (). i i i v v v Fig. (): Forwrd relizion of 2-conrol offoli ge using he NCV librry Definiion 3: A ful olern circui is circui h cn coninue un-inerruped performnce when fuls re developed in he circui. Fig. (b): Ful olern design of qunum ge V/V Fig. (c): Ful olern design of Fig.() using Clifford group i i Fig. (d): Ful olern NNC free design of Fig.(c) using nive pproch

2 A ful-olern qunum circui is such h, where Phse ges () in he circui is free from conrol nodes. Using Clifford group, designing such circui is possible. In he Clifford group, ges such s dmrd ge (), Puli's Z ge, Phse ge nd CNO ge re included. Besides h, lso so-clled / ges re ddiionlly included. he ful olern design of V/V ge is shown in Fig. (b). By mpping ech of he qunum ges presen in n MC srucure, n equivlen ful olern design cn be obined. he equivlen ful olern design of Fig. () is depiced in Fig. (c). ome well-known merics h re considered o mesure performnce of ful olern design re sed below. Definiion 4: he qunum cos (QC) of ge (g) is he number of elemenry qunum operions performed o design he specified ge (g). Definiion 5: he -coun vlue of circui is he number of / ge used o relize he inpu circui. Definiion 6: he moun of ime h is required o obin he firs oupu fer pplying n inpu o ge is ermed s - cycle. Definiion 7: he minimum number of -cycles h re required o execue ll operions in ful olern circui is ermed s deph. Definiion 8: he number of ges in he circui is clled ge coun. Definiion 9: Number of cycles used o execue he enire circui is ermed s circui deph. Definiion 8: he Neres Neighbor Cos (NNC) of qunum circui is defined s he sum of he NNCs of is ges; where for qunum ge (g) hving conrol c h line nd rge h line hs he NNC vlue c. An NNC free circui lwys hs eiher -qubi or 2-qubi ges in is configurion which operes on djcen qubis by producing 0 Neres Neighbor Cos. ABLE : QUANUM GAE AND EIR PROPERIE he NNC free design of Fig. () using nive pproch is given in Fig. (d). B. Reled Work nd Conribuion hough Clifford group [8, 9] of operors hs he biliy o produce ful-olern circuis, i is no universl operor se nd, herefore, non-clifford ge is dded o he exising Clifford group nd new ful olern group Clifford is formed. everl design nd uomion pproches for consrucing ful olern circuis using Clifford group re repored in [20-23], where he reserchers hve minly sudied he minimizion of circui size ( coun, Ge coun) nd run ime prmeers (circui deph, deph). In recen work [2], he uhors hve shown heurisic pproch o design ful olern MC srucures using he Clifford group. Bu he uhors hve resriced heir work up-o 3-conrol MC ges due o he incresing complexiy of he pproch. Moreover, for he physicl relizion of hese ful-olern qunum circuis, NNC free designs of such circuis re noher imporn requiremen which is no sudied in [2]. he mos common wy of convering qunum circui o NNC free one, is by insering WAP ges. owever, in order o minimize he overll cos of he circui, pproprie WAP sequences hve o be incorpored nd o chieve furher minimizion in he NNC free design, pos synhesis schemes [24-28] hve been proven very efficien. In [29], emple mching echnique is proposed, which replces he WAP inserion scheme nd repors suible resuls wih respec o NNC reducion. Alhough he presened scheme works for Clifford circuis, i hs no been pplied o he MC mppings proposed in [2]. In his work, we show h n improved mpping of MC ges o NNC free Clifford circui is possible. ype (symbol) NO (X) Mrix Digrm Properies X 0 X 0 ype (symbol) Puli-Y Mrix Digrm Properies 0 i i 0 Y Y.Y I Y 0 0 Y -i 0 CNO (C) Conrolled V (V N) dmrd () ge i i i i ou c Puli s Z 0 Z.Z I Z 0 0 Z - Conrolled 0 i 0 i V V V - V V V (V N) 2 2 I V 0 i i ) ge e iπ/2 e iπ /2 0 - ) e iπ /4 e iπ /4 Phse ge ( Z) Z iπ /2 Z.V. 0 e V.Z.; ge 0 0 e iπ/4 e iπ /4 Phse ge iπ /2. V..W.; 0 e W.. ge Conrolled ge 0 0 e iπ /2 e iπ / i Phse ge Conrolled ge 0 e iπ/2. V..W. W.. 0 i

3 his is chieved by se of emples for higher order MC ges (we demonsre his for up o 5-conrol lines) nd NNC reducion by considering he liner neres neighbor (LNN) propery for CNO ges. Our combined pproch improves he circui performnce significnly. he res of he pper is orgnized s follows. he proposed echnique wih exmples is discussed in ecion II. ecion III dels wih he experimenl resuls nd finl concluding remrks pper in ecion IV. II. PROPOED ECNIQUE ere we discuss he improved design mehodology for consrucing NNC free nd ful olern MC srucures. he enire design procedure is composed of wo phses. In he firs phse, we hve modified he lgorihm of [2] nd hve incorpored circui opimizion procedure iming for beer mpping of MC ges o Clifford srucures. Firs, we execue heurisic emple mching scheme over he qunum circuis, followed by wo seps of phse ge reordering nd redundn ges cncellion which ogeher improve our design in comprison o he exising pproch [2]. In he nex phse, we hve resrucured he designs o mke hem NNC free. Phse : Improve mpping scheme for ful olern designs of NCV bsed MC ges: Before sing he enire scheme, here we re defining emple which is exensively in our proposed mpping procedure. emple : he defined emple (s shown in Fig. 2()) is composed of pir of offoli ges in which boh he ges hve he sme conrol lines ( nd ) h c on he sme rge line (). Mpping of ech of he offoli ge o is ful olern srucure does no led o n opimizion, nd for h we hve chosen sregy o mp boh he offoli ges joinly ino ful olern one. We hve ppended he redundn CNO ges o he iniil srucure in Fig. 2(b) nd hen find is opimized NCV-bsed qunum circui fer elimining he redundn ges s depiced in Fig. 2(c). hen we mp ech V/V ges o is equivlen Clifford group bsed subcircuis. Nex, we pply he possible rnsformion rules h hve been defined in ble 2 over he circui followed by cncellion of redundn ges presen in he circui. Fig. 2(): A pir of 2-conrol offoli ge Fig. 2(b): Afer ppending redundn ges in Fig. 2() V V V V V V Fig. 2(c): Equivlen NCV ge librry bsed relizion of Fig. 2() he inermedie design is presened in Fig. 2(d). he finl ful olern srucure corresponding o Fig. 2() is obined nd hs been shown in Fig. 2(e). Now we presen he proposed mpping scheme, which is composed of seven seps: ep: ke n MC ge (G) ou of which ful-olern design is o be obined. ep2: Relize his ge s n NCV qunum ge circui using he pproch inroduced in [4]. ep3: he obined circui conins elemenry qunum ges like NO, CNO, V/V nd offoli ges. Now, scn he circui from lef o righ nd replce ll he V/V ges wih is equivlen Clifford group bsed sub-circuis. ep4: Reorder he circui s per he rules sed in ble 2. ep5: Nex, subsiue ll he conrolled-phse ges () exising in he circui wih equivlen conrol free Phse ges. Now, he circui is composed of dmrd ges (), Clifford /, CNOs ges nd pir of wo-conrol offoli ges. ep6: As he presen circui conins pir of 2-conrol offoli ge, i is necessry o replce h srucure wih n equivlen Clifford-bsed ful olern sub srucure (emple). o his end, subsiue he sub-circui s in Fig. 2() presen in he circui wih he equivlen ful olern sub srucure of Fig. 2(e). ep7: Apply he following wo circui opimizion rules in he circui: ep7.: As he produc of wo -ges is he ideniy remove wo consecuive -ges from he srucure. ep7.2: Move phse ges (/ ) cross he conrol node of CNO ges such h -deph minimizes bu no movemens of phse ges re llowed cross he rge line of NO or CNO ges. Oherwise, he funcionliy of he enire circui will ge modified. ep7.3: Inerchnge he conrol line nd rge line of wo qubi phse ges when i is necessry ccording o rules defined in ble 2. Fig. 2(e): Clifford bsed ful olern relizion of Fig. 2() fer opimizion ABLE 2: OME IMPORAN RANFORMAION Fig. 2(d): Reordering nd hen removing exrneous ges from he circui For beer undersnding of he enire design procedure, consider he following exmple, where he improved ful olern design of 3-conrol offoli ge is discussed. Exmple : Iniilly, he 3-conrol offoli ge which is depiced in Fig. 3() is decomposed using he mpping echnique sed in [4]; leding o he circui in Fig. 3(b). Now, by replcing ll he V/V ges o is equivlen Clifford group bsed sub-circuis, he resuling srucure shown in Fig. 3(c) is derived. Nex, we reorder he phse ges in Fig. 3(c)

4 nd obin he modified design in Fig. 3(d). hen, replcing ll he Phse ges () in Fig. 3(d) wih equivlen Clifford bsed ful olern sub-circuis yields he circui given in Fig. 3(e). While reviewing he circui of Fig. 3(e), we found subsrucure h mches he emple in Fig. 2. o, subsiue he sub-srucure wih emple nd ll he redundncies re removed from he design s per he principle sed in ep7. Finlly, in Fig. 3(f), he obined nd improved ful olern design of he hree conrol offoli ge is shown. V V V V Fig. 3(): 3-conrol Fig. 3(b): NCV bsed Fig. 3(c): Clifford group offoli ge wih one decomposiion of Fig. 3() bsed decomposiion of ncillry line Fig. 3() Fig. 3(d): Reordering of phse ges in Fig.3(c) Fig. 3(e): Replcemen of ll qunum / ges wih conrol free phse ges Fig. 3(f): Clifford bsed ful olern relizion of Fig. 3() fer opimizion Phse 2: Mking Clifford srucures NNC free: Above, we hve shown how o mke qunum circui ful olern using Clifford group ges nd hve lso formuled he sregies o improve he cos merics of h circui. In his phse, we process he circuis h hve been obined fer execuing sep5 from he previous secion in order o mke hem NNC free. As sed in ecion I, here lredy exis severl well-known echniques o mke ny rbirry circui NNC free. Mos of he pproches pply emple mching schemes followed by locl nd globl reordering scheme of ges o mke he circui NNC free. Bu s hese pproches need some serious compuion, he complexiy of his pproch quickly increses. ere we re showing n esier wy o mke our ful olern circuis NNC free using wo simple emples h replce he non-nnc free ges or sub-circuis wih equivlen NNC-free sub-srucures. Ineresingly, i cn be noed h using ny of he well-known NNC opimizion echniques for he fulolern circuis considered here would yield he sme resul. owever, in order o reduce he complexiy, we hve used he defined emples nd ssocied sregy s proposed in his work. As he presen circui conins ges, / ges, CNO ges nd pir of wo conrol offoli ges in is design, where ll he ges nd / ges in he circui minin physicl djcency beween he wo neighboring lines. h mens hey ll re NNC free srucures. Bu he CNO ges nd he pir of offoli ges h re presen in he design re no NNC free srucures nd we hve o mke hem NNC free o rein he enire design s NNC zero. o his end, we re uilizing he following emples: emple2: A CNO ge hving one ncillry line hs o be replced wih he emple s defined in Fig. 4. c Fig. 4: A CNO ge nd is NNC free design emple3: he NNC free ful-olern design of pir of ncill free 2-conrol offoli ges is given in Fig. 5. Iniil digrm of he offoli ge is depiced in Fig. 5(). Afer going hrough hree inermedie sges (Fig. 5(b) o Fig. 5(d)), he finl design is obined in Fig. 5(e). Bu if he pir of 2-conrol offoli ges conins ncillry lines, n for exmple, hen he srucure will be bi differen. For his scenrio, firs we hve o decompose he iniil srucure using he previously defined emple2 so h he decomposed design becomes NNC free. hough exernlly his decomposed design looks NNC free, i is no s i conins pir of ncill free 2-conrol offoli ges which re no NNC free. o, we hve o subsiue his pir of 2-conrol offoli ges wih he NNC free ful-olern srucure h we hve defined erlier (in Fig. 5). he bove sed ide is explined in Fig. 6, where he iniil design of pir of 2-conrol offoli ges conining n ncillry lines nd is decomposed srucure re depiced in Fig. 6() nd Fig. 6(b), respecively. Afer subsiuing he pir of ncill free 2-conrol offoli ges (shown in he doed box) in Fig. 6(b) wih NNC free ful olern srucure of Fig. 5, he finl NNC free ful olern design of Fig. 6() is obined in Fig. 6(c). V V V V V V Fig. 5(): A pir Fig. 5(b): of 2-conrol Appending wo C- Fig. 5(c) Equivlen NCV offoli ge NO ge in Fig. 5() relizion of Fig. 5(b) Fig. 5(d): Replcing ll qunum V/V ge by is equivlen Ful olern srucure Fig. 6(): Pir of 2- conrol offoli ges wih n ncillry lines Fig. 5(e): NNC free srucure of Fig. 5() Fig. 6(b): NNC free decomposiion of Fig. 6()

5 . Fig. 6(c): Opimized NNC free srucure of Fig. 6 () If he defined emples re mpped over he ful olern design of hree conrols offoli ge in Fig. 3(e), hen he circui will be rnsformed ino NNC free design nd fer pplying opimizion rules over i finlly Fig. 7 is obined. Exmple 2: he NNC free ful olern srucure of 5- conrol MC ge is shown in Fig. 8. he iniil design of he MC ge is depiced in Fig. 8(). Fig. 8(b) presens he ful olern relizion of he ge. Afer removing ll he non NNC bsed ges from he circui, he finl ful-olern NNC free design is given in Fig. 8(c). Fig. 7: NNC free ful olern design of 3-conrol MC ge c 4 c 3 Fig. 8(): 5-conrol MC ge Fig. 8(b): Equivlen ful olern design of Fig. 8() c 4 c Fig. 8(c): NNC free ful olern design of 5-conrol MC ge

6 III. EXPERIMENAL REUL ere we give wo ses of resuls in wo sepre bles ble 3 nd ble 4 where we hve shown vrious cos merics fer pplying our enire pproch over MC ges nd benchmrk circuis [30]. We hve compred our resuls wih [2] where ful olern circuis re designed followed by NNC free echnique s sed in [29]. Percenge improvemens in boh he bles re clculed nd re shown. In ble 3, i cn be observed h improvemens re ined when MC sizes increses bu in h ble for he firs wo rows improvemens re no seen s he ges in he respecive wo rows do no hve ny ncillry lines. No. of conrol lines Benchmrk nmes ABLE 3: FAUL-OLERAN NNC FREE DEIGN OF MC GAE Cos merics obined using echnique [2] followed by [29] C: -Coun GC: Ge Coun n: No. of conrol lines coun coun CONCLUION IV. Proposed design cos merics % Improvemens n QC C GC CD QC C GC CD QC% CD% Averge improvemen 7 4 ABLE 4: COMPARAIVE ANALYI UING BENCMARK CIRCUI Cos merics obined using echnique [5] followed by [2] Proposed design cos merics % Improvemens n QC C GC CD QC C GC CD QC CD mod5dder permnen 2x min\2of5\d esign# bnhmrk_ bnhmrk_ bnhmrk_ QC: Qunum Cos CD: Circui Deph coun Averge improvemen 7 In his work, we hve proposed n improved design echnique for consrucing NNC free ful olern qunum circuis relying on he Clifford librry for MC ges. he proposed scheme hs wo phses. In he iniil phse, we hve improved he mpping scheme from MC ges o is fulolern design by reducing ge coun nd circui deph, while in he second phse, we hve mde hese circuis NNC free. In fuure work, we will invesige how o relize higher order MC ges s NNC free ful olern qunum circuis. REFERENCE [] P. W. hor, Algorihms for qunum compuion: discree logrihms nd fcoring, Foundions of Compuer cience, pp , (994). [2] L. K. Grover, A fs qunum mechnicl lgorihm for dbse serch, in heory of compuing, pp , (996). [3] hp://dl.cm.org/ciion.cfm?id [4] hp://dvnces.sciencemg.org/conen//3/e bsrc [5] hp:// [6]. ffner e l. clble mulipricle ennglemen of rpped ions, Nure, pp , (2005). [7] M. Lfores e l., Using error correcion o deermine he noise model, Physicl Review A, pp , (2007). [8] B. Kne, A silicon-bsed nucler spin qunum compuer, Nure, pp , (998). [9] J. Ghosh, J. e l. igh-fideliy CZ ge for resonor bsed superconducing qunum compuers. Phys. Rev. A 87, (203). [0] M. Veldhors e l., A wo Qubi Logic Ge in ilicon, Nure, (204) [] R. Brends, e l., Logic ges he surfce code hreshold: uperconducing qubis poised for ful-olern qunum compuing, Nure pp (204). [2] P. Boykin,. Mor, M. Pulver, V. Roychowdhury, nd F. Vn, A new universl nd ful-olern qunum bsis, Informion Processing Leers, vol. 75, no. 3, pp. 0 07, (2000). [3]. Buhrmn, e l., New limis on ful olern qunum compuion, In Proceedings of he 47 h Annul IEEE ymposium on Foundions of Compuer cience, pp. 4 49, (2006). [4] D. Miller, R. Wille, nd Z. snin, Elemenry qunum ge relizions for muliple-conrol offolli ges, In Proc. In'l ymp. on Muliple-vlued Logic,pges pp , (20). [5] Z. snin nd D. Miller, rnsforming MC Circuis o NCVW Circuis, pringer's Lecure Noes in Compuer cience., pges 77-88, (20). [6] A. Brenco e l., Elemenry ges for qunum compuion, AP Physicl Review, pp , (995) [7] Z. snin, R. Wille, nd D. M. Miller, Relizing reversible circuis using new clss of qunum ges, In Proc. Design Auomion Conf., pp. 36-4, (202). [8] D. Goesmn, bilizer codes nd qunum error correcion, rxiv preprin qun-ph/ , (997). [9] P. Aliferis, D. Goesmn, nd J. Preskill, Qunum ccurcy hreshold for concened disnce-3 codes, Qunum Info. Compu., vol. 6, pp , (2006). [20] M. Amy, D. Mslov, M. Mosc, Polynomil-ime -deph Opimizion of Clifford circuis vi Mroid priioning, 6 h Inernionl Conference on Reversible Compuion, Jpn (204) [2] D. Miller, M. oeken, R. Drechsler, Mpping NCV Circuis o Opimized Clifford Circuis, Reversible Compuion (204). [22] M. Amy, D.Mslov, M. Mosc, M. Roeeler, A mee-in-he-middle lgorihm for fs synhesis of deph-opiml qunum circuis, IEEE rns. on CAD 32(6), pp (203) [23] P. elinger, Qunum circuis of -deph one, Phys. Rev. A 87, (203) [24] M. eedi, R. Wille, nd R. Drechsler, ynhesis of qunum circuis for liner neres neighbor rchiecures, Qunum Informion Processing, vol. 0, no. 3, pp , (20). [25] R. Wille, A. Lye, nd R. Drechsler. Exc Reordering of Circui Lines for Neres Neighbor Qunum Archiecures. IEEE rnscions on Compuer Aided Design of Inegred Circuis nd ysems, 204 [26] A. Chkrbri nd. ur-koly, Neres neighbour bsed synhesis of qunum boolen circuis, Engineering Leers, pp , December (2007). [27] M.. A. Khn, Cos reducion in neres neighbour bsed synhesis of qunum boolen circuis, Engineering Leers, pp. 5, (2008). [28] Y. ir, M. Nknishi,. Ymshi, nd Y. Nkshim, An efficien mehod o conver rbirry qunum circuis o ones on liner neres neighbor rchiecure, In Inernionl Conference on Qunum, Nno nd Micro echnologies, pp , (2009). [29] M. Rhmn nd G. Dueck, ynhesis of Liner Neres Neighbor Qunum Circuis, 0 h Inernionl Workshop on Boolen Problems, Germny (202),. [30] hp://