Quantum data bus in dipolar coupled nuclear spin qubits

Transcription

1 Quntum dt bus in dipolr coupled nucler spin qubits Jingfu Zhng, 1 Michel Ditty, 1 Dniel Burgrth, 2 Colm A. Ryn, 1 C. M. Chndrshekr, 1,3 Mrtin Lforest, 1 Osm Mouss, 1 Jonthn Bugh, 1 nd Rymond Lflmme 1,3 1 Institute for Quntum Computing nd Deprtment of Physics, University of Wterloo, Wterloo, Ontrio, Cnd N2L 3G1 2 IMS nd QOLS, Imperil College, London SW7 2BK, United Kingdom 3 Perimeter Institute for Theoreticl Physics, Wterloo, Ontrio, Cnd N2J 2W9 Received 30 Mrch 2009; published 16 July 2009 We implement n itertive quntum stte trnsfer exploiting the nturl dipolr couplings in spin chin of liquid-crystl NMR system. During ech itertion, finite prt of the mplitude of the stte is trnsferred nd, by pplying n externl opertion on only the lst two spins, the trnsferred stte is mde to ccumulte on the spin t the end point. The trnsfer fidelity reches one symptoticlly through incresing the number of itertions. We lso implement the inverted version of the scheme which cn trnsfer n rbitrry stte from the end point to ny other position of the chin nd entngle ny pir of spins in the chin, cting s full quntum dt bus. DOI: /PhysRevA PACS number s : Lx, k I. INTRODUCTION In quntum computtion nd quntum communiction, the trnsfer of n rbitrry quntum stte from one qubit to nother is fundmentl element. The most obvious method to implement the quntum stte trnsfer QST on n rry of qubits is bsed on sequence of SWAP gtes for neighboring spins. In spin qubit systems, the SWAP gte up to known phse fctor cn be implemented through the evolution of the dipolr coupling between the neighboring spins for 1/ 2D time by decoupling the other spins, where D denotes the dipolr coupling strength. In experiments, however, the required decoupling opertions re hrd to implement if the spins cnnot be individully ddressed by spectrl selectivity, e.g., in lrge-size solid-stte NMR systems. This mkes the direct implementtion of such gtes in lrge spin system chllenging. To overcome this problem, schemes bsed on lwys on spin systems were proposed 1,2. The stte cn be trnsferred with unit fidelity in engineered spin chins or networks with XY interctions 3. However, the required finetuned XY couplings re not found in nturl spin systems 4. In other schemes bsed on spin chins with Heisenberg interctions 1,5 or with double-quntum Hmiltonin 4, the fidelity of the QST cnnot pproch unity in sclble systems. The bove limittions cn be relxed significntly by pplying gte opertions to receive nd store the trnsferred stte 6,7. The gtes re only pplied to two spins t one end of spin chin. In this pper, we experimentlly implement the QST in liquid-crystl NMR system bsed on this scheme. Opposed to previous experimentl implementtions 7 where the required XY interctions were engineered by rdio-frequency pulses nd sclr couplings, the dipolr couplings exist nturlly in the system nd re directly exploited for the QST. The dipolr couplings re much stronger up to 2 to 3 orders of mgnitude thn the sclr couplings nd therefore cn significntly speed up the implementtion of the logicl gtes for quntum informtion processing 8. The trnsfer with high fidelity is chieved in n itertive mnner. Ech itertion trnsfers finite prt of the input mplitude to the trget spin t the end of the chin. The fidelity of the trnsfer symptoticlly pproches unity by incresing the number of itertions. We lso experimentlly demonstrte the time-inverted version of 6. Through this, full quntum dt bus is implemented, where rbitrry unknown quntum sttes cn be steered to ny position of the chin. This is lso useful for the selective excittion of one spin, which is ddressed by the two-spin gtes, rther thn by its individul properties, e.g., chemicl shift in NMR. As opposed to previous schemes 9, globl control is not required. Surprisingly, the reversl opertion cn lso be used to entngle ny pir of spins in the chin by opertions t its end only. We demonstrte the entngling opertion in the qubits t the end points of the chin. The QST nd its reversl opertions men tht the chin is relly used s wire with n input, n output, nd no gtes in the middle; the mny-body Hmiltonin of the chin is responsible for the trnsport. Only two spins t the end re required to ddress. The fidelity of trnsfer converges exponentilly fst to unity with respect to the number of itertions. The required number of itertions to chieve good fidelity e.g., lrger thn scles roughly linerly with the system size 6. Moreover, this method is stble when the engineered Hmiltonin in implementtion devites the required Hmiltonin 10. Hence, our method scles fvorbly with the size of the spin chins nd suitble for lrge-size systems, such s solid- or liquid-crystl NMR systems, where the differences of the chemicl shifts re too smll to ddress ll the spins individully. II. ITERATIVE TRANSFER ALGORITHM IN A SPIN CHAIN Our first gol is to trnsfer the stte 0 1 from spins j to N in n N-spin chin. The Hmiltonin for spins 1 to N 1 is represented s /2009/80 1 / The Americn Physicl Society

2 ZHANG et l. H = 1 2 N 1 j,k=1;k j D jk 2 z j z k x j x k y j y k, where j x, j j y, nd z denote the Puli mtrices with j indicting the ffected spin. Noting tht H preserves the totl number of excited spins 1,3, we hve U 0 = e i 0, N 1 U j = k k, 3 k=1 where U =e i H nd e i is the 1,1 mtrix element of U. The stte 0 denotes ll spins pointing up, nd j denotes ll spins up except the spin j pointing down. The min opertion is the two-spin gte pplied only on spins N 1 nd N, nd, in itertion n, the gte is denoted s d n c n 0 W c n,d n = I 1,2,...,N 2 1, 4 0 c n d n N 1,N where I 1,...,N 2 denotes the unit opertor for spins 1 to N 2. The bsis order for spins N 1 nd N is 00, 01, 10, nd 11. Noting c n 2 d n 2 =1, one finds W c n,d n c n N 1 d n N = N. The N spin system is initilized into the input stte 0 j by setting spin j in the system to stte 0 1. Here, j is the loction of the sender receiver of the QST for the inverse protocol, which is on some rbitrry spin of the quntum dt bus. It is sufficient to only discuss the trnsfer of j becuse U only introduces known phse fctor before 0 see Eq. 2 nd W c n,d n does not chnge 0 see Eq. 4. Itertion n is represented s Q j,n = I 1,...,N 2 W c n,d n U I N. After n itertions, one obtins n = T j,n j = A k,n k, using Eqs Here T j,n =Q j,n...q j,2 Q j,1, A N 1,n =0, nd where N k=1 A N,n = pn, p n = p n 1 N 1 U I N n 1 2, with p 0 =0 nd 0 = j. W c n,d n is obtined by setting d n = e i pn 1 / pn, c n = N 1 U I N n 1 / pn p n p n or C n b Itertion n 10 2 FIG. 1. Color online The numericl simultion solid nd experimentl results dt mrked by for the probbility p n of the QST s function of itertions in the four spin system used in experiments see text for dipolr couplings for trnsferring stte from spins 1 to 4 nd entngling the two spins b when =2.1 ms. In b, p n cn be pproximted s the observble coherence C n dot dshed, see Eq. 19 where C n p n when n 2. The experimentl dt cn be fitted s 0.65p n nd 0.77C n shown s the dshed curves, respectively. In strict nerest-neighbor chins, it cn be shown 6 tht p n converges to unity by incresing the number of itertions. In the present cse, we hve lso non-nerest-neighbor interctions, but numericl results show p n still pproches unity, with convergence speed which depends on the evolution time see Fig. 1. The process of QST fter lrge number of itertions cn be presented s T j,n 0 j e in 0 N, 12 i.e., spin N ends with the stte e in 0 1 nd e in is known. We cn exploit the inversion of T j,n to implement the QST from spin N to spin j, i.e., without pplying the externl opertion directly on the spin j to evolve it into stte 0 1. Hence, the spin chin functions s quntum dt bus, which cn trnsfer rbitrry unknown sttes to ny qubit. This method lso llows to crete selective excittion tht does not require spectrl selectivity, e.g., chemicl shift in NMR, to ddress spin j. The externl opertions re only pplied to spins N 1 nd N. By tking the inner product of Eq. 7 with N nd using Eq. 8 one obtins p n = j T 1 j,n N 2, 13 i.e., p n is the fidelity for generting j by pplying T 1 n,j to N. The cretion of the selective excittion for spin j is represented s T 1 j,n 0 N e in 0 j. 14 By modifying the input stte, one cn obtin T 1 j,n e in 0 N 0 j 11. The method of the inverse QST, furthermore, cn be used to entngle rbitrry spins j,k indirectly by cting t spins N 1 nd N only. This cn be done by designing pulse nlogously to Eq. 6, nd the required pulse sequence is very similr to the inverse QST. For this purpose, we set the input stte s n entngled stte of pir of spins j nd k represented s

3 QUANTUM DATA BUS IN DIPOLAR COUPLED NUCLEAR jk = j k / Itertion n cn still be represented s Eq. 6, where Q j,n is rewritten s Q j,k,n, noting tht it depends on the input stte. W c n,d n is obtined in similr wy by chnging p 0 = N jk 2 nd 0 = jk. After lrge number of itertions, we obtin T j,k,n jk N, 16 where T j,k,n =Q j,k,n...q j,k,2 Q j,k,1. From Eq. 16 one cn entngle spins j nd k with high fidelity by pplying T j,k,n on 1 N, represented s T 1 j,k,n N jk. 17 The fidelity for generting jk is lso represented by Eq. 13 through replcing T 1 j,n by T 1 j,k,n, nd j by jk. The numericl simultion for p n is illustrted s Fig. 1 b. III. EXPERIMENTAL RESULTS We use the four protons in orthochlorobromobenzene C 6 H 4 ClBr dissolved in the liquid-crystl solvent ZLI-1132 s four qubits to implement the experiments. The Hmiltonin is represented s 4 H NMR = i i z 1 2 i=1 4 k=2,j k D jk 2 z j z k x j x k y j y k. 18 Through fitting the spectr 12 obtined by Cory48 13 nd one-dimensionl MREV-8 pulse sequences nd referring to the spectr of molecules with similr structures 8,14, we mesure 1 =106.2, 2 = 187.7, 3 = 58.6, nd 4 =91.3 with respect to the trnsmitter frequency, D 12 = , D 13 = 149.4, D 14 = 93.2, D 23 = 716.0, D 24 = 236.6, D 34 = Hz, nd the effective trnsverse relxtion times T 2 s 91, 87, 88, nd 82 ms 15. The NMR spectrum obtined by Cory48 from the therml equilibrium stte th = 4 i i=1 z is shown in Fig. 2. All experiments strt with the devition density mtrix ini = , which cn be prepred by the double-quntum coherence Hmiltonin 16 H d = k=2,j kd d jk j x k x j y k y in molecule with C 2v symmetry 17. However, we choose to generte the effective H d using grdient scent pulse engineering pulse 18. Using temporl verging, we prepre ini by summing the three sttes U d th U d, U d th U d,2 th, where U d =e it dh d by choosing d t d =8.00/D In the numericl simultion, we prepre ini with fidelity of 99.97%. Figure 2 b shows the NMR spectrum obtined by collective /2 pulse in the experiment when the system lies in ini. The NMR peks mrked by indicte the single-quntum trnsitions between mgnetic quntum numbers 2 nd 1. We demonstrte the QST by trnsferring 0 from spins 1 to 4 by choosing 0 = x, y, z, nd I, respectively. Becuse T j,n is spin preserving, the trnsitions mrked by in Fig. 2 b cn represent the QST strting with the input stte We therefore cn ignore in ini b FIG. 2. Color online NMR spectr thick obtined by Cory48 pulse sequence from the therml equilibrium stte nd by b collective /2 pulse from ini. The thin spectr show the results by simultion. The plot s verticl xes hve rbitrry units. The NMR peks mrked by indicte the single-quntum trnsitions between mgnetic quntum numbers 2 nd 1. nd omit the negtive frequency spectrl region. The input stte is prepred by pplying n opertion U ini to ini. With incresing n, T 1,n trnsforms to symptoticlly, where =e in z/2 0 e in z/2. In experiments, we removed the phse fctor between nd 0 by phse correction. For fixed n, we implement the unitry T 1,n U ini using one GRAPE pulse. The experimentl results of the QST fter 100 itertions for the vrious input sttes re shown s Figs. 3 3 d, respectively. b c d FIG. 3. Color online d NMR spectr for implementing the QST from spins 1 to 4 fter 100 itertions, when the input sttes re chosen s x , y , z , nd I , respectively, where the redout opertion e i 4 is pplied to obtin observble signls in c or d. The plot s verticl xes hve the sme scle

4 ZHANG et l FIG. 4. Color online NMR spectr for implementing the selective excittion nd quntum dt bus for spin 2. Exploiting the trnsformtion between the computtionl bsis nd energy eigenbsis nd ignoring the difference of T 2 of the four protons, we cn pproximtively obtin A k,n in Eq. 7 through mesuring the mplitudes of the peks mrked by in Figs. 3 3 c by choosing the signls in Fig. 2 b s the reference. Therefore, we obtin p n = A 4,n 2. For the input sttes x , y , nd z , p 100 is mesured s , , nd , respectively. All other A k,100 2 re below To observe p n incresing with n, we lso implement the QST by choosing vrious n when the input stte is x The mesured p n is shown in Fig. 1 s the dt mrked by, which cn be fitted s 0.65p n. Next we implement the selective excittion or quntum dt bus for spin 2. The reverse QST strts with the input stte x obtined by pplying R 4 y =e i 4 y /4 to ini. When n=100, T 1 2,n trnsforms x to with probbility close to 1, where =e in z/2 x e in z/2. The experimentl results re shown in Fig. 4. The fidelity of excittion is mesured s We choose 14 = 1 4 / 2 s the trget to demonstrte the entngling opertion in spins 1 nd 4. To mesure the fidelity, we rewrite Eq. 13 s p n = 0000 n 2 19 by replcing j by 14. Here n = P 1 T 1,4,n 4 where P denotes the opertion to prepre 14 from 0000 e.g., see 20. When p n is close to 1, we cn obtin p n pproximtely by pplying redout opertion e i 1 to n. Noting tht in ini does not contribute observble signls for mesuring p n, we pproximte p n s the coherence C n = Tr n, 19 where n =U tot,n ini U tot,n with U tot,n =e i 1 P 1 T 1,4,n e i 4 x /2. The simulted nd mesured C n is shown in Fig. 1 b. The FIG. 5. Color online NMR spectr for mesuring the fidelity of the genertion of 1 4 / 2. experimentl dt cn be fitted s 0.77C n. Figure 5 illustrtes the NMR spectr when n=8. The opertions U d, U d, R 4 y, T 1,n U ini, T 1 2,n R 4 y, nd U tot,n re experimentlly implemented using the GRAPE pulses with fidelities in theory lrger thn 0.99, respectively. The pulse lengths re 10 ms for U d nd U d, 20 ms for the other pulses. The experimentl errors could minly result from the inhomogeneities of the mgnetic field, imperfect implementtion of GRAPE pulses, nd decoherence. In order to estimte the qulity of the experimentl spectr, we lso list the idel ones in simultion shown s the red thin curves in Figs IV. CONCLUSION We hve given n NMR implementtion for vrious importnt tsks of quntum control tht, in principle, cn be chieved indirectly by controlling the end of spin chin. The dipolr couplings nturlly existing in the liquid-crystl NMR system re directly exploited for the QST. The experimentl results demonstrte the successful control of the spin system with dipolr couplings by the GRAPE pulses. First, we implemented the trnsfer of n rbitrry quntum stte. Second, by implementing the reverse QST, we hve creted full quntum dt bus which is controlled by the two-qubit end gtes. Finlly, s nother ppliction of the reverse QST, we proposed nd demonstrted different method to implement n entngling opertion. ACKNOWLEDGMENTS We thnk D. Cory, C. Rmnthn, S. Bose, G. B. Furmn, nd T. S. Mhesh for helpful discussions. We cknowledge support by the EPSRC under Grnt No. EP/ F043678/1, NSERC, Quntum Works, nd CIFAR. 1 S. Bose, Phys. Rev. Lett. 91, S. Bose, Contemp. Phys. 48, ; D. Burgrth, Ph.D. thesis, University College London, M. Christndl, N. Dtt, A. Ekert, nd A. J. Lndhl, Phys. Rev. Lett. 92, ; C. Di Frnco, M. Pternostro, nd M. S. Kim, ibid. 101, P. Cppellro, C. Rmnthn, nd D. G. Cory, Phys. Rev. Lett. 99, E. B. Fel dmn nd A. I. Zenchuk, Phys. Lett. A 373, D. Burgrth, V. Giovnnetti, nd S. Bose, Phys. Rev. A 75, J. Zhng, N. Rjendrn, X. Peng, nd D. Suter, Phys. Rev. A 76,

5 QUANTUM DATA BUS IN DIPOLAR COUPLED NUCLEAR 8 T. S. Mhesh et l., Curr. Sci. 85, ; T. S. Mhesh nd D. Suter, Phys. Rev. A 74, J. Fitzsimons nd J. Twmley, Phys. Rev. Lett. 97, ; J. Fitzsimons, L. Xio, S. C. Benjmin, nd J. A. Jones, ibid. 99, ; P. Cppellro, C. Rmnthn, nd D. G. Cory, Phys. Rev. A 76, D. Burgrth, Eur. Phys. J. Spec. Top. 151, In our experimentl system, we directly implemented the bckwrd time evolution U of Q 1 j,n by forwrd evolution of H. This is not necessry 2, but it simplified the presenttion. Actully, one cn design Q j,n in Eq. 6 for the reverse QST using H. Hence, in the implementtion of Q 1 j,n the evolution is still U. 12 H. Tkeuchi et l., Chem. Lett. 29, D. G. Cory, J. B. Miller, nd A. N. Grrowy, J. Mgn. Reson , M. K. Henry, C. Rmnthn, J. S. Hodges, C. A. Ryn, M. J. Ditty, R. Lflmme, nd D. G. Cory, Phys. Rev. Lett. 99, Hmiltonin tomogrphy for lrge-size liquid-crystl NMR systems is still hrd problem currently. New theoreticl nd experimentl techniques re developing, e.g., L. D. Field, Annu. Rep. NMR Spectrosc. 59, ; B. Bishy nd N. Suryprksh, J. Phys. Chem. A 111, ; D. Burgrth, K. Mruym, nd F. Nori, Phys. Rev. A 79, R J. Bum et l., J. Chem. Phys. 83, ; J.-S. Lee nd A. K. Khitrin, Phys. Rev. A 70, ; J. Chem. Phys. 121, G. B. Furmn, J. Phys. A 39, J. Bugh et l., Phys. Cn. 63, ; N. Khnej et l., J. Mgn. Reson. 172, ; C. A. Ryn, C. Negrevergne, M. Lforest, E. Knill, nd R. Lflmme, Phys. Rev. A 78, J. Zhng, F. M. Cucchietti, C. M. Chndrshekr, M. Lforest, C. A. Ryn, M. Ditty, A. Hubbrd, J. K. Gmble, nd R. Lflmme, Phys. Rev. A 79, I. L. Chung et l., Proc. R. Soc. London, Ser. A 454,