Supplementary Figure S1 Characterization of the two-qubit system. a-d, When the nuclear spin is polarized along the

Transcription

1 Suppleeary Figure S Characeriaio of he wo-qubi syse. a-d Whe he uclear spi is polaried alog he axis is free precessio sigal abou he axis is odulaed by he relaive agle bewee he codiioal local fields. a-b Whe 9 he cere lies of he free precessio sigals fro he saes ad are offse fro each oher ad he agle ca be easured. The sybols are experieal daa ad he lies are fiig o a cosie fucio. c-d Whe is ued o be 9 by adjusig he orieaio of he ageic field he cere lies of he free precessio sigals coicide. The sregh was easured o be. MH by fiig he oscillaio sigal. e-f Whe he uclear spi is polaried alog he axis is free precessio abou he axis gives.() MH i agreee wih he pulsed opically deeced ageic resoace easuree.

2 Suppleeary Figure S Sae oography of he wo qubis iiially i a basis sae. The syse was prepared i wo basis saes ad he sae oography was carried ou before (a & b) ad afer (c & d) applyig he desiged corolled-not (C e NOT ) gae. The iagiary pars of he easured desiy arices were egligible (o displayed). a Sae oography o he iiial sae wih sae fideliy of.98(). b Sae oography o he iiial sae shows ha he sae is prepared wih sae fideliy of.89(). c Sae oography o he resula sae afer apply he gae o.the uclear spi was lef uchaged wih a fial sae fideliy of.9(). d Sae oography o he resula sae afer applyig he gae o. The uclear spi was flipped wih a fial sae fideliy of.8().

3 Suppleeary Table S Tiig paraeers (i s) of he DD sequeces desiged for realiig various wo-qubi quau gaes. N is he uber of pulses ad deoes he ierval bewee he pulses (oe ha he pulse sequeces are syeric so oly he firs halves of he DD sequeces are lised). a Corolled-NOT gae (C e NOT ) N b Nuclear spi Hadaard gae (H ) N c Nuclear spi Pauli-X gae (X ) N

4 d Nuclear spi Pauli-Z gae (Z ) N e Two-qubi NULL gae (NULL) N

5 Suppleeary Noe : Theoreical descripio of he NV cere syse Pure-dephasig approxiaio Here we discuss he pure-dephasig approxiaio for he NV cere syse. We reark ha such a approxiaio is o ecessary for he desig proocol o be applicable hough i does siplify he uerical opiiaio i his work. I his subsecio we use he ageic uber o label he cere spi saes. We odel he syse by cosiderig a NV cere coupled o he bah spis hrough hyperfie ieracio. The bah cosiss of he N hos uclear spi ad C uclear spis wih aural abudace of.%. Uder a exeral ageic field B Bˆ B he syse Hailoia ca be wrie as (N) ( i) ( i) ( i e (N) ( i) ( i) ( i) ( ( S ) BS P ( I ) BI Sα I I β I (S) where ẑ is alog he NV syery axis.8 GH is he ero-field spliig of he NV cere spi e.8 MH/G ad are respecively he gyroageic raios of he cere spi S ad he ih uclear spi I ( uclear spi ad. kh/g for he iroge spi) (N). kh/g for a C α represes he hyperfie couplig bewee he elecro spi ad he ih uclear spi β ( i represes (N) he dipolar ieracio bewee he ih ad jh bah spis P.9 MH is he quadrupole oe of he N spi ad suaio over i ad j is iplied. Noe ha he subscrip ad superscrip (N) is reserved o refer o he N hos uclear spi. Whe ad he hyperfie ieracio sregh i are boh eb e B he direc flippig of he NV cere spi bewee he saes ad he sae is suppressed by he large ero-field spliig. Therefore is a good quau uber ad perurbaio reae ca be perfored by separaig he full syse (i.e. cere spi + uclear spis) Hailoia by e ( i) ( i) (N) ( i α I( i) B I( i) ( (N)) I( i) β I( S B S S P I ( i) * ( i)* e α B( i) e α I( i) S B S B S S Bx iby x i y whereα ˆ x B α α x y α S S is ad are defied by coparig he correspodig ers. Clearly he cere spi eigesaes are all eigesaes of he uperurbed Hailoia wih eigeeergies give by E ad E eb. Perurbaio reae up o secod order gives he effecive codiioal Hailoia (acig o he uclear spis) (S)

6 h eb E (S) where is a projecio operaor. Evaluaig we have h ( i) ( i) ( i e B α I( i) BI( i) I( i) β I( E ( i h BI( i) I( i) β I(. E E Sraighforward copuaio gives (N) (N) ( i) ( i) ( (N)) BI( i) BδG I( i) h E C P I ( i ( i ( i) ( I ( β δβ ) I wih he secod order correcios give by C δg δβ e E (N) (N) (N) E xy ( i) ( i) ( i) ( i) x y x y ( i) ( i) x y ( i) ( i) ( i) xx xy x e ( i) ( i) ( i) yx yy y E ( i B B ( I ) α where we have assued α α E ( i) ( α α E xy produc bewee vecors. We reark ha i i i g by g ( ) δg ( ) / (). (N) ( i) ( i) ( i) for C δg & δβ (S) (S) (S) (N) α o be diagoal 9 ad here refers o he Kroecker δg is relaed o he ehaced g-esor The essece of hese codiioal Hailoias is capured by wriig geerically i he for of he uclear spis ( () e i /( ) ( i h ( i ) ( i ) ( j ) ω I I β I. Geerally he quaiaio axes of ) deped o he sae of he cere spi ( ). Sice is of order for a oderaely srog hyperfie couplig he ehacee of he g-esor of he uclear spi due o virual flip-flops of he cere spi is sigifica whe he ageic field is o aliged alog he NV syery axis 9. This g-esor ehacee however eers hrough ω ad ω i reversed sig. This iplies ha uder a geeral ageic field he quaiaio axes () ω i of a paricular uclear spi do o coicide. We also reark ha becoes a

7 well-defied quaiaio axis for he ih uclear spi whe alos diagoal which are he codiios uilied i ref.. B B ad α is Codiioal evoluio of he arge uclear spi To illusrae he codiioal evoluio of he uclear qubi we resric ourselves o he syse fored oly by he NV cere spi ad he arge C spi (idexed by ()). Whe he syse is allowed o evolve freely he syse propagaor wihi he U( ) exp ih where for pure-dephasig approxiaio is give by breviy we se hroughou our reae. I geeral ˆ ˆ ( () () () deoig he ui vecor alog ω ) so h ad h do o coue ha is he codiioal uclear spi evoluio represes precessio abou differe axes which suffices o geerae uiversal uclear spi operaors. Whe he syse is subjeced o a series of cere spi -pulses wih iig iervals{ } { N} he syse propagaor is give by U{ } u{ } where { } ih N ih ih u e e e wih or for N beig eve or odd ad u is siilarly defied. () ˆ

8 Suppleeary Mehods Deeriaio of ω ad ω The codiioal evoluio of he uclear qubi ca be revealed by sudyig is free precessio whe he cere spi is prepared i differe saes. To be specific we wrie ω si x cos ad ω ad deoe he spi operaors for he uclear qubi by I. Noe ha i his oaio x x ˆ ad ˆ i geeral. To icorporae he effec of icoplee polariaio we suppose he uclear qubi is iiially represeed by he desiy arix. We furher deoe he polariaio of he arge spi righ afer he polariaio procedure by Tr( I ). Whe a high degree of polariaio is achieved i.e. he off diagoal elees of has agiude precessio sigal due o he off-diagoal ers ca be egleced. Therefore he expecaio value of I () as a fucio of free evoluio ie is give by ad so he coribuio of he free I ( ) [cos cos( )si ]. (S) Clearly he sregh is give by he frequecy of he free precessio sigal. By coparig he correspodig sigal wih polariaio alog ad he polariaio ad he relaive agle ca be easured. I Suppleeary Fig. S we show he free precessio sigal whe he ageic field (fixed a G) was orieaed alog differe direcios. Suppleeary Figs. Sa-b show he sceario whe B was aliged a ~ fro he NV syery axis. The agle was deeried fro he free precessio sigal i Suppleeary Fig. Sb o be. Suppleeary Figs. Sc-d show he correspodig sceario whe B was aliged a ~ fro he NV axis which is he seig for he experieal resuls icluded i he ai ex. The syeric oscillaio abou he coo cere lie of he precessio sigals idicaes ha was ued o be 9. A siilar se of procedures would also reveal iforaio of ω (Suppleeary Figs. Se-f). Nuerical siulaio of wo-qubi gaes Regardig he wo coupled spi-/'s as a -level syse he average fideliy F of execuig a uiary gae G by a syse propagaor U { } ca be siplified by where Tr( UO ju GO jg ) d j d Tr F U U G G (S8) dd ( ) d ad j O is a coplee orhooral operaor basis for he -level syse saisfyig Tr( OO). The DD sequeces have he syeric fors j k jk { N } for N pulses ad { N N } for N pulses. 8

9 We adoped a uerical axiiaio proocol based o a schee siilar o he oe described i he GRAPE algorih. Sarig wih a iiial guess of{ } i each axiiaio sep we updae by F such ha F where is a sall uerical paraeer. F ca be calculaed hrough he sraighforward evaluaio of U which requires oly he kowledge of he ier-qubi ieracio bu o he deailed specru of he bah. As discussed i ref. such a approach guaraees covergece oly o local bu o global axia of F. Besides give resources of N -pulses i is desirable o also iiie such ha he coherece is beer proeced ad he gae speed is faser. Therefore oe has o choose a appropriae iiial guess { } for a proper desig of he DD gae sequece. I paricular we firs surveyed a special subse of N -pulses sequeces wih he for od( ) such ha he paraeer space is oly wo-diesioal. I addiio we resric values of ad such ha he oal gae operaio ie is reasoably shor. The iiial guess { } is he deeried by he values of ad such ha F aais a global axiu over his resriced wo-diesioal paraeer space. The DD gae sequece { } DD is he foud by uerical opiiaio of F i he larger paraeer space. We prese i Suppleeary Table S a coprehesive lis of he desiged DD gae sequeces discussed i Fig. of he ai ex. 9