2. The general decomposition of single-qubit, two-qubits and three-qubits quantum gates

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1 NESL QNT CCT FO -QT QNT GTE: POGLE QNT GTE PLO ENÍCO DE SOS d ENS N OS Dprtmto d Eghr d Tlformátc vrsdd Fdrl do Crá - DET/FC C.P. 67 Cmpus do Pc Fortl-C rsl Qutum computto hs ttrd much ttto, mog othr thgs, du to ts pottlts to solv clsscl NP prolms poloml tm. For ths rso, thr hs growg trst to uld qutum computr. O of th sc stps s to mplmt th qutum crcut l to rl gv utr oprto. Ths ts hs solvd usg dcomposto of utr mtrcs smplr os tll rch qutum crcuts hvg ol sgl-quts d CNOTs gts. sull th gol s to fd th mml qutum crcut l to solv gv prolm. ths ppr w go dffrt dro. W propos grl qutum crcut l to mplmt spcfc qutum crcut just sttg corrl th prmtrs. othr words, w propos progrmml qutum crcut. Ths ops th posslt to costru rl qutum computr whr svrl dffrt qutum oprtos c rld th sm hrdwr. Th cofgurto s proposd d ts optcl mplmtto s dscussd. Kwords: Qutum computto, qutum crcuts, dcomposto of utr mtrcs. troduo Qutum computto hs ttrd much ttto du to ts pottlts to solv clsscl NP prolms poloml tm. Th most fmous mpl s th Shor s forto lgorthm []. Sc Shor s lgorthm proposl, much ffort hs rld to dsg d costru qutum computrs. tll, qutum crcuts hv proposd to solv som spcfcs prolms, s Dutsch prolm [], qutum Fourr trsform [,] d othrs. Th f tht qutum gt c ult usg ol sgl-qut d CNOT gts [] courg th rsrchrs to loo for mthod to fd th qutum crcut tht rls gv qutum gt rprstd utr mtr. Two dcompostos hs usd for ths ts, th cos-s dcomposto [6] d Crt s KK dcomposto [7,8]. O th othr hd, qutum crcut dsgg usg dffrt ds of gtc lgorthm hs succssfull rportd [9-]. Th m dsdvtg of such lgorthms s ts vlo of covrgc for lrg crcuts. Howvr, th rsult otd tds to optmd. Hc, t s possl, owds, to fd th qutum crcut tht rls gt, usg group thor thorms or rtfcl tllgc. sull, th m s to fd th mml qutum crcut l to solv gv prolm. ths wor w go dffrt dro. Our m s to propos grl qutum crcut l to mplmt spcfc qutum crcut, just sttg corrl th prmtrs. othr words, w propos progrmml qutum crcut. Ths ops th posslt to costru qutum computr whr svrl dffrt qutum oprtos c rld th sm hrdwr. Sc th uvrsl qutum crcut proposd th prmtrs r chgd sttg corrl clsscl vrls s voltg d currt, t s possl to us clsscl computr to cotrol th uvrsl qutum crcut. Ths ops th posslt to rl svrl srvcs for qutum formto s qutum cotrol d slf-cofgurl crcuts mog othrs. Ths wor s outld s follow: So th grl dcomposto of sgl-qut, two-qut d thr-qut gts r rvwd. So th uvrsl qutum crcut for -qut qutum gt s prstd. So possl optcl mplmtto s dscussd. So som pplctos r dscussd d, t lst, th coclusos r prstd So 6.. Th grl dcomposto of sgl-qut, two-quts d thr-quts qutum gts sgl-qut gt c dcomposd s:

2 P... d Sous d.. mos () () cos s s cos whr r th Pul mtrcs. Ths dcomposto s wll ow from lght polrto thor, sc polrto chg c rld usg two rtrdrs d o polrto rottor tw thm. From () w s tht grl sgl-qut gt s prmtrd thr vrls. two-qut gt c dcomposd s []: p D D ϕ ϕ ϕ ϕ ϕ. (),, d r sgl-qut qutum gts whl D s o-forl two-qut gt rsposl for th o-locl chrrstc of th gt. (), r g th Pul mtrcs d th qutum stts,,, form th wll ow mgc ss: Ψ Ψ. ordr to fd th prmtrs of th dcomposto ()-() lt us tll cosdr th followg mtrcs [7]: Λ () () () (6) (7) (8) (9)

3 vrsl qutum crcut for -qut qutum gt: progrmml qutum gt f s th utr mtr whos dcomposto o s loog for, th th followg mtrcs c dfd: Now, o c fd th grld sgulr vlu dcomposto of th mtrcs d : SX CX ()-(),, d X r utr mtrcs whl C d S r ogtv dgol mtrcs. Sc d r oth rl d smmtrc mtrcs d, furthrmor,, th utr mtrcs d r rltd : F F whr F s dgol mtr whos lmts ssum ol th vlus d -. Now, worg wth ()- () o ots: F X FS C Th mtr CFS s dgol utr mtr, hc, t s wrtt s: FS C d th gls ()-() r th otd from [7]: Λ t lst, th locl oprto,, d c otd from () () () () () () (6) (7) (8)

4 P... d Sous d.. mos (7) X usg som trvl lgr. Hc, grl two-qut gt s prmtrd fft vrls (thr for th o-locl oprto d twlv for th locls oprtos). sg ()-(), t ws show [] tht two-qut gt c costrud usg thr CNOT gts d ght sgl-qut gts, s show Fg. : Th sgl-quts gts Fg. r rltd to th prmtrs of ()-() : - - π whr s th dtt mtr. Hc, gv two-qut gt, o c us (8)-() to fd ts dcomposto d, ftr, usg ()-() to fd th quvlt qutum crcut usg ol sgl-qut gts d CNOTs. thr-qut gt C c dcomposd s []: C d c c N N N p p () d (,,,) r, rspvl, two-qut d sgl-qut gts. Th qutum crcuts for N, d C r show Fgs, d, rspvl []. (9) () () () () () Fg. Two-qut gt costrud usg ght sgl-qut gts d thr CNOTs. (8) (6) ()

5 vrsl qutum crcut for -qut qutum gt: progrmml qutum gt H s th Hdmrd gt whl th othr sgl-qut gts r: c π π Th sgl-qut gts r: 6 d c π π 8 c N H H Fg. Qutum crcut for utr mtr N. 8 cd H H 6 Fg. Qutum crcut for utr mtr. (8) (9) () () 8 C N N Fg. Qutum crcut for grl thr-qut qutum gt.

6 P... d Sous d.. mos Hc, grl thr-qut gt hs 8 vrls.. uvrsl qutum crcut for -qut qutum gt Now, w propos grl qutum crcut pttr tht c usd to mplmt -qut qutum gt. Th d s to crt sc cll d qutum gt c ult l th clls. Th cll for grl -qut qutum gt s show Fg.. Fg. vrsl cll for -qut gt. s c s from Fg., th sc cll s composd () sgl-qut gts d (-) CNOTs, hc, cll hs ()(-) vrls. For ch CNOT thr s vrl dctg f t s vtd or dvtd (worg s utr gt). lthough w hv ot provdd mthmtcl proof tht -qut gt c ult usg th schm of Fg., t s ot dffcult to otc tht ths s tru. Th schm of Fg. c mplmt sgl-qut gt of th quts of th qutum us. t c lso mplmt th CNOT tw two quts of th qutum us. Hc, th smplst cs, ch cll c rprst ol sgl-qut gt or CNOT tw two quts (ll th othr gts r st s dtt) d svrl clls c usd to mplmt qutum gt. s mpls, o c sl osrv tht two-qut gt c ult usg two clls of th tp: Fg. 6 vrsl cll for two-qut gts. whl thr-qut gt c ult usg lss th squcs of th followg cll.

7 vrsl qutum crcut for -qut qutum gt: progrmml qutum gt Fg. 7 vrsl cll for thr-quts qutum gts. sg 7 clls of th tp show Fg. 7 o c costru th qutum gts N d show Fgs. d, rspvl. Th uvrsl qutum crcut provds th posslt of mplmtto of qutum vrso of th FPG (Fld Progrmml Gt rr) tcholog, whch prmts th hrdwr g st v softwr. f, ssmlg som sc clls for -qut qutum crcut, s show Fg., svrl dffrt qutum oprtos c mplmtd just djustg th prmtrs of th sgl-qut gts d lg or ot th CNOTs gts. Wh CNOT gt s ot ld, t wll wor s dtt gt.. possl optcl mplmtto of th progrmml qutum gt ordr to mplmt progrmml qutum gt, o must l to costru sgl-qut gts wth sl djustl prmtrs d CNOT gts tht c swtchd to dtt gt. Hr, w wll us lght polrto s th qut. For ths qutum sstm, s dscussd for, sgl-qut gt c ult usg two rtrdrs (or compstors) d polrto rottor tw thm. Such dvc c costrud usg optcl frs cols or rottl wv-plts [ d rfrcs thr ]. ths lst, vlus for rtrdrs d polrto rottor c chvd rottg th wv-plts (ths chgs th dcomposto of th cdt fld th fst d low s of th rfrgt crstls). Hc, prcpl, djustl sgl-qut gt c mplmtd usg mcromchcl djustmt of th rotto of th wv-plts. Ths mthod s ot good o cus t prmts ol slow vrto of th prmtrs. Howvr, polrto modultors hv costrud sd o ul - smcoduor wvgud mcrostruurs, hvg pottll smll s d hgh spd modulto [6]. Hc, full d sl djustl sgl-qut gts for polrto codd quts sm to prolm tht c solvd th r futur. much hrdr prolm s th optcl mplmtto of CNOT gts. umr of solutos hv proposd [7-]. Hr, w wll dscuss th usfulss of th soluto prstd [-] th progrmml qutum gt. Th st-up proposd thos rfrcs, s show Fg. 8. D out out D PS H HH D PS PS PS H D C α β H T H c c t t Fg. 8 CNOT gt mplmtd wth polrto m splttrs (PS) d two-photo tgld stt. PS H : PS horotl-vrtcl ss. PS : PS dgol ss (π/, π/). D - r sgl-photo dtors.

8 P... d Sous d.. mos Th optcl stup show Fg. 8 mplmts th CNOT gt ol wh sgl-photo s dtd D or D d othr sgl-photo s dtd D or D. Th output stt of th stup Fg 8 s: Ψ f [, H, H,, ] Ψ [ α HH β H α β H ] [ α HH β H α β H ] ( XZX) [ α H β HH α H β ] [ X] [ α H β HH α H β ] ( XZX) [ ] u [ ][ X] () () () () (6) () Ψ u s th uslss prt tht cots th stutos whr o or two photos wr dtd th D - d/or D -. Furthr,,H() ms sgl-photo gog to D d othr sgl-photo gog to D (D ), whl -,H() ms sgl-photo gog to D d othr sgl-photo gog to D (D ). Osrvg () d () o ss tht th succss prolt of CNOT oprto s /6. Howvr, f o uss sgl-qut oprtos to corr th output stt ccordg to whr th dtos wr otd (dtos D d D XZX th cotrol qut, dtos D d D X th trgt qut d dtos D d D XZX th cotrol qut d X th trgt qut) th prolt of succss gos to /. crucl compot th CNOT mplmtto of Fg. 8 s th tgld pr of photos. f std of ( HH )/ / o hd usd ( H H )/ /, o would gt th utr oprto (X) CNOT (X), tht s, CNOT tht vrts th trgt wh th cotrol qut s H. O th othr hd, f o uss th dstgld stt ( HH H )/ / std of th ll stt, th stup Fg. 8 mplmts th dtt oprto. Hc, th progrmml uvrsl gt c costrud usg cotrolld CNOTs s show Fg. 8, ut hvg th posslt to choos from ( HH )/ / d ( HH H )/ /. f th CNOT hs to vtd, th two-photo stt ( HH )/ / s usd, othrws, th stt ( HH H )/ / s usd. Th m prolms wth CNOT mplmtto of Fg. 8 r th csst of rll tgld two-photo sourcs d ts prolstc hvor. Ths lst s th prc to pd for trg to mplmt o-lr oprto usg lr dvcs. Howvr, t hs show [] tht th o-lrt of qutum msurmts c usd to mplmt (lmost) dtrmstc qutum gts. r dtrmst CNOT usg qutum o-dmolto msurmt hs proposd [].. pplctos of th progrmml qutum gt Th frst pplcto s th costruo of th qutum prsor s dlmm gm [-7], for two plrs, usg two clls of th uvrsl qutum crcut for two-qut gt. Th m ojv of qutum gm, o-cooprtv gm, s to chv optml qulrum (Prto) stutos whr th clsscl strtgs c ot ol th Nsh s qulrum (whr oth plrs r ot courgd to chg thr dcsos). Th m d s to g dvtg ovr clsscl oppot, usg of qutum strtgs. f, wh th clsscl scro s cosdrd, plr c vlut ol o ltrtv d t o sgl dcso (ormll rrvrsl). ut, wh th trsc prlllsm sd qutum strtgs (usg th suprposto of qutum stts) s usd, th plr c cosdr multpl ltrtvs smultousl. of ths dsrd proprts c osrvd usg th prsor dlmm, tht s dscrd short s follows: f w cosdr tht two popl r ccusd to commt som crm d thr s o posslt to rl commucto tw thm, whch chrrs o-cooprtv gm,

9 vrsl qutum crcut for -qut qutum gt: progrmml qutum gt th hv two possl strtgs: cooprt (C) or df (D). Th followg tl shows th possl os d thr poffs for oth plrs [-7 d rfrcs thr ]. lc \ o C D C (,) (,) D (,) (,) Tl Tl of poffs for prsor s dlmm. th clsscl scro, ch plr s ot supposd to ow tht th dvrsr c cooprt (whch mpls rs), so th commo dcso s to df. Ths stuto th Nsh qulrum s mportt cus thr s o ctv for chgg ths dcso wthout rug rs. Hc, thr s o rtol ssumpto whch th c gt optml fl stuto th Prto optml whch oth would cooprt d, togthr, th could rch th st stuto. Now, cosdrg th qutum scro, th plrs dcsos r mplmtd through qutum gts. Th strtgs to cooprt d df r rprstd th qutum stts C d D, rspvl. Th tl totl stt s ssumd to th tsor produ CC. Th totl stt s procssd qutum gt J tht tgls oth quts, J CC cos(γ/) CC s(γ/) DD. Th d of solutos otd dpds o th mout of tglmt crtd [6]. Th strtgs r mplmtd, ch plr dvdull, pplg sgl-qut qutum gt. Th stup s llustrtd Fg. 9. J J Fg. 9 Qutum crcut for qutum prsors dlmm. Th qutum crcut of Fg. 9 c sl mplmtd usg two clls of th uvrsl qutum crcut for two-qut gt, s show Fg.. Cll Cll Fg. Qutum prsor s dlmm mplmtto usg clls of th uvrsl qutum crcut for two-qut gt. Fg., for cll, p( γ ), d whl for cll, ( ). Hc, th gm c pld chgg γ, tht cotrols th mout of tglmt crtd, d chgg th strtgs through of th vrto of d. Now, usg two clls of th uvrsl qutum crcut for thr-qut gt, o c costru, for mpl, Toffol gt, s show Fg.

10 P... d Sous d.. mos Cll Cll Fg. Toffol gt costrud from two clls of th uvrsl qutum crcut for thr-qut gt. For Cll p( π ) d for Cll p( π ), p( π ). O th othr hd, th Toffoll gt oprtd lvl std of s costrud s show Fg.. Cll Cll Fg. Toffol gt, oprtg th lvl, costrud from two uvrsl clls for thr-quts qutum gts. Hc, o c chg from Toffol gt oprtg lvl to Toffol gt oprtg lvl just chgg th prmtrs of sgl-qut gts. 6. Coclusos W proposd progrmml qutum crcut, tht s, grl qutum crcut, sd o sglqut gts d CNOTs, l to mplmt -qut qutum gt, just sttg th prmtrs corrl. Such prmtrs, for optcl sgl-qut gts, r th compstors d polrto rottor prmtrs, tht s, th prmtrs of polrto modultor, whch c costrud usg movl wv-plts cotrolld mcromchcll or smcoduors wvguds. th cs of CNOTs, th prmtr s sgl cotrol tht vts or ot th CNOT. f th CNOT s dvtd t wors s dtt mtr. mpl usg optcl CNOT gt ws prstd. t lst, som mpls of th us of th uvrsl qutum crcut proposd wr prstd: t ws show th mplmtto of qutum gm of two plrs (th prsor s dllm) d th mplmtto of th Toffol gt oprtd lvl d. cowldgmts Ths wor ws supportd th rl gc FNCP.

11 vrsl qutum crcut for -qut qutum gt: progrmml qutum gt frcs. P. W. Shor (997), Poloml-tm lgorthms for prm forto d dscrt logrthms o qutum computr, S J. Comp., 6, L. Chug d Y. Ymmoto (99), smpl qutum computr, qut-ph/9.... Nls,. L. Chug, Qutum Computto d Qutum formto, Cmrdg v. Prss, Cmrdg, Egld, ()... rco (996), Qutum phscs d computrs, Cotmp. Phs., 7,, rco, C. H. t,. Clv, D. P. Dco, N. rgolus, P. Shor, T. Sltor, J.. Smol d H. Wfurtr (99), Elmtr gts for qutum computto, Phs. v.,,, öttö, J. J. rt,. rgholm d.. Slom (), vrsl qutum computto, qutph/89. (cos-s dcomposto) 7... Tucc (), troduo to Crt s KK dcomposto for QC progrmmrs, qut-ph/ Y. Njm, Y. Kwo d H. Sgw (), w lgorthm for producg qutum crcuts usg KK dcomposto, qut-ph/ P. ust (), Evolvg qutum crcuts usg gtc progrmmg, Joh. Ko, dtor, Gtc lgorthms d Gtc Progrmmg t Stford. Stford oostor, sr.j.c.com/.html.. L. Spor, H. rum, H. J. rst, d N. Swm (999), Qutum computg pplctos of gtc progrmmg, dv. G. Progr.,.. T. Yu d H. (), Gtc lgorthm form qutum crcut dsg volvg smplr tlportto crcut. Tchcl rport, sr.j.c.com/86.html... Krus d J.. Crc (), Optml crto of tglmt usg two-qut gt, Phs. v., 6, pp. 69/-8.. G. dl d C.. Dwso (), uvrsl qutum crcut for two-qut trsformtos wth thr CNOTs gts, qut-ph/777.. F. t d C. P. Wllms (), lto of grl thr-qut qutum gt, qut-ph/78.. D. Drcso (Edtor) (998), Fr Optc Tst d surmt, Prtc-Hll, Nw Jrs. 6. N. Y. Gordv, K. J. Gordo d G. S. ullr (), Tul lro-optc polrto modultor for qutum dstruto pplctos, Opt. Comm., E. Kll,. Lflmm d G. J. lur (), schm for ffct qutum computto wth lr optcs, Ntur, 9, T.. Ptm,. C. Jcos d J. D. Frso (), Eprmtl dmostrto of qutum crcut usg lr optcs gts, qut-ph/9 (). 9. F.. Spdlr, H. L, d J. P. Dowlg (), Hgh-fdlt lr optcl qutum computg wth polrto codg, qut-ph/8.. T. C. lph,. G. Wht, W. J. uro, d G. J. lur, Smpl schm for ffct lr optcs qutum gts, Phs. v., 6, () /-6.. T.. Pttm,. C. Jcos, d J. D. Frso (), Prolstc qutum logc oprtos usg polrg m splttrs, Phs. v., 6, pp. 6/-9.. T.. Pttm,. J. Ftch,. C. Jcos, d J. D. Frso (), Eprmtl cotrolld-not logc for sgl photos th cocdc ss, Phs. v., 68, pp. 6/-.. S. Schl, K. Nmoto, W. J. uro, d P. L. Kght (), surmt-ducd olrt lr optcs, qut-ph/8.. K. Nmoto, d W. J. uro (), Nr dtrmstc lr optcl cotrolld-not gt, Phs. v. Ltt., 9, pp. /-.. J.Esrt,. Wls d. Lwst (999), Qutum gms d qutum strtgs, Phs. v. Ltt., 8,, J. Du, H. L, X. Xu,. Sh, J. Wu, X. Zhou d. H (), Eprmtl rlto of qutum gms o qutum computr, Phs. v. Ltt., 88,, 79/-. 7. L. Zhou d L-. Kug (), Proposl for optcll rlg qutum gm, Phs. Ltt.,, 6-.

Quantum Circuits. School on Quantum Day 1, Lesson 5 16:00-17:00, March 22, 2005 Eisuke Abe

Quantum Circuits. School on Quantum Day 1, Lesson 5 16:00-17:00, March 22, 2005 Eisuke Abe Qutum Crcuts School o Qutum Computg @Ygm D, Lsso 5 6:-7:, Mrch, 5 Esuk Ab Dprtmt of Appl Phscs Phsco-Iformtcs, CEST-JST, Ko vrst Outl Bloch sphr rprstto otto gts vrslt proof A rbtrr cotroll- gt c b mplmt