Suppressing environmental noise in quantum computation through pulse control

Transcription

1 Ocober 999 Ž. Physics Leers A Suppressing environmenl noise in qunum compuion hrough pulse conrol Lu-Ming Dun ), Gung-Cn Guo Deprmen of Physics nd lborory of Qunum communicion, Qunum Compuion, UniÕersiy of Science, Technology of Chin, Hefei , PR Chin Received 9 July 999; cceped 22 Augus 999 Communiced by P.R. Hollnd Absrc A scheme bsed on pulse conrol is described for suppressing noise in qunum compuion wihou he cos of sringen qunum compuing resources. I is shown h environmen-induced noise nd decoherence, wheher in qunum memory or in ge operions, ll cn be much reduced by pplying suible sequence of bi-flipping nd phse-flipping pulses. q 999 Published by Elsevier Science B.V. All righs reserved. PACS: Hk; Dd; p Qunum compuion hs become very cive field ever since he discovery h qunum compuers cn be much more powerful hn heir clssicl counerprs w 3 x. Qunum compuers c s sophisiced qunum informion processors, in which clculions re mde by he conrolled ime evoluion of se of coupled wo-level qunum sysems Ž qubis.. Coherence in he evoluion is essenil for king dvnge of qunum prllelism. However, here is mjor obscle o he relizion of qunum compuion. Decoherence of he qubis cused by he inevible inercion wih noisy environmen will mke qunum informion oo frgile o be of ny prcicl use. Recenly, ineres in qunum compuion hs incresed drmiclly becuse of wo respecs of dvnces owrd overcoming he bove ) Corresponding uhor. E-mil: lmdun@vorex.uibk.c. E-mil: gcguo@usc.edu.cn difficuly. Firs, combinion of series of innovive discoveries, such s qunum error correcing codes w4,5 x, ful-olern error correcion echniques wx 6, nd concened coding wx 7, hs yielded he imporn hreshold resul w7,8 x, which promises h rbirrily ccure qunum compuion is possible provided h he error per operion is below hreshold vlue. Hence, noise below cerin level is no n obscle o relible qunum compuion. Second, gre progress hs been mde owrd buildw9 x. Some ing he necessry qunum hrdwres simple bu rel qunum compuions hve been demonsred in bulk spin resonnce qunum sysw2,3 x; nd rdicl scheme, using semicon- ems ducor physics o mnipule nucler spins, hs been recenly proposed, which indices promising roue o lrge-scle qunum compuion w4 x. Ful-olern qunum error correcion schemes re effecive only when he error re per operion is below hreshold vlue. Operions in qunum r99r$ - see fron mer q 999 Published by Elsevier Science B.V. All righs reserved. Ž. PII: S

2 40 ( ) L.-M. Dun, G.-C. GuorPhysics Leers A compuers include rnsmission or sorge of qunum ses nd qunum logic. For qunum logic, he y6 esimed hreshold error re is bou 0 w7,8 x. Wih he presen echnology, he rel noise seems necessrily beyond his hreshold, even for he mos promising qunum hrdwre wx 8. Therefore, i is essenilly imporn o furher suppress noise before king he procedure of ful-olern qunum error correcion. Here, we propose noise suppression scheme which operes by pplying sequence of bi-flipping nd phse-flipping pulses on he sysem, wih he pulse period less hn he noise correlion ime. I is shown h noise in he pulse-conrolled sysem cn be grely reduced. The scheme bsed on pulse conrol is relively esy o implemen in experimens. In fc, clever pulse mehods Ž clled refocusing echniques. hve been developed for yers in nucler mgneic resonnce specroscopy Ž NMR. o effecively remove mny kinds of inercions mong he spins h re considered unwned or unineresing w5 x. In rew6 x, Viol nd Lloyd used his echnique cen work o comb environmenl noise in compuer memory wih specific single-qubi dephsing model. Ler, he echnique ws immediely exended o more generl cses w7,8 x, nd generl heory for suppressing memory noise hs been developed in he group conex w9,20 x. In his pper, we consider he generl ype of environmenl noise, including clssicl or qunum dephsing nd dissipion s is specil cses. By pplying suible pulse sequences, i is found h he noise in he conrolled sysem is much reduced; nd furhermore, he scheme cn be redily exended o suppress noise in qunum ges Žreferring o wo-bi or muli-bi qunum ges, whose error res re much lrger hn h of single-qubi roions w9,4,2 x.. Suppose pg is he error re per operion, nd p0 is he ddiionl error re inroduced by ech pulse. Our resul shows h he error re per operion in he pulse-conrolled sysem pproximely reduces o p p r, g 0 dec c where dec nd c re respecively he decoherence ime nd he noise correlion ime. Since p is normlly very smll, he reducing fcor p r is 0 dec c less hn even h he noise correlion ime is considerbly smller hn he decoherence ime. Therefore, by his scheme i is possible for he reduced error re o in he hreshold vlue, sy, 0 0 y6. This is desirous resul for relible qunum compuion. The only requiremen in our scheme is h he pulses should be pplied frequenly so h he pulse period is less hn he noise correlion ime. The scheme coss no ddiionl qunum compuing resources. Firs we show how o use pulse conrol o comb decoherence of single qubi in qunum memory. The qubi is described by Puli s operor s. In he inercion picure, he mos generl form of he Hmilonin describing single-qubi decoherence due o environmenl noise cn be expressed s Žseing "s. Ž. Ž. Ž. HI s s G, sx, y, z where GŽ., generlly dependen of ime, re noise erms, which my be clssicl sochsic vribles or sochsic qunum operors, corresponding clssicl noise or qunum noise, respecively. For environmenl noise in qunum compuers, i is resonble o ssume h GŽ. Ž sx, y, z. sisfy he condiions ² G Ž. : s 0 nd ² : env G Ž. G Ž. sf Ž j., Ž 2. b env b where ² PPP : env denoes verge over he environ- men. In Eq. Ž. 2, ll fbž j. re correlion funcions nd jsžy. r c. The quniy c chrcerizes he order of mgniude of he noise correlion imes. Differen noise correlion erms my hve differen correlion imes. Bu we ssume h hey hve he sme order of mgniude, which is denoed by c. For simpliciy, in he following we direcly cll c he noise correlion ime. Suppose h he qubi is iniilly in pure se C Ž 0.:. Under he Hmilonin Ž., fer shor ime i evolves ino mixed se rž. fer king verge over he environmen. The difference beween he ses rž. nd C Ž 0. : snds for errors. The error re p cn be described by psyfž., where FŽ. is he inpu-oupu se fideliy, defined s FŽ. s² C Ž 0. rž. C Ž 0.:. From Eqs. Ž. nd Ž. 2, i is no difficul o obin n explici perurbive expression for he error re. Up o he second order of he Hmilonin, he resul is b 2 ps2 ² Ds Ds : shh fb d2d, Ž 3.,b ž 0 0 c

3 ( ) L.-M. Dun, G.-C. GuorPhysics Leers A where Ds s s y² s : s, nd ² PPP : s denoes v- erge over he sysem. Eq. Ž. 3 is derived wih pure inpu se. However, i remins rue when he qubi is iniilly in mixed se. In his cse, we define he error re by psyfež., where FeŽ. is he ennglemen fideliy w22 x, nurl exension of he inpu-oupu se fideliy o he mixed se circumsnce. Wih his definiion, he expression for he error re remins compleely sme s Eq. Ž. 3. Now we show how o use pulse conrol o reduce he error re in qunum memory. We pply sequence of bi-flipping nd phse-flipping pulses on he qubi, wih he pulse period D nd pulse widh w. I is required h w<d so h he environ- men-induced sysem evoluion during he shor ime w is negligible. A ime s0, no pulse is pplied, nd is operion is represened by he uni operor I. A ime s, we begin o pply in urn he bi-flipping nd he phse-flipping pulses. The pulse-induced operions re hus respecively I,s x,s z,s x,s z, PPP. Four pulse periods mke up conrol period. Le U 2 denoe 2 T½exp yih HIŽ. d 5, where T PPP 4 indices h ime-ordered produc is ken in he brcke. In he Ž nq. h conrol period Žform ime 4nD o ime 4Ž n q.., he effecive sysem evoluion under pulse conrol is represened by he following evoluion operor: z Ž 4nq4. D x Ž4 nq3.d nq Ž 4nq3. D Ž4 nq2.d U ss U s U =s z U Ž4 nq2. s x U Ž4 nq. ½ Ž Ž4 nq.d 4 nd syt exp yi H s H IŽ. s d Ž 4 nq4. z z 4 nq3. yi H s H IŽ. s d Ž 4 nq3. y y Ž 4 nq2. yi H s H IŽ. s d Ž 4 nq2. x x Ž 4 nq. Ž 4 nq. H D Ž. I 5 4n yi H d ½ syt exp yi H 4Ž nq. 4n 5 sx, y, z s G Ž. d. Ž 4. The effecive noise erms G Ž 4nD q. wih 0F- 4 in Eq. Ž. 4 re defined s follows: D ž 3 D j D D 4 js0 4 G Ž 4n q. s h G 4n qj q, Ž 5. where he 3=4 coefficien mrix y y hs wh jxs y y. Ž 6. y y Afer pulse conrol, he only difference in he sysem evoluion Ž. 4 is h he noise erms GŽ. re replced by he corresponding effecive noise erms G. Form Eqs. Ž. 2 nd Ž. Ž. 5, i is no difficul o ge he correlions of he effecive noise G Ž.. Then, subsiuing hese correlions ino Eq. Ž. 3, we obin he error re pc fer pulse conrol. The finl resul is ž c 2 pc D s, Ž 7. p where c is he noise correlion ime, nd is dimensionless ner-o- fcor Žis explici form cn be found in w7 x., which is unimporn o our resul nd will be omied in he following discussion. From Eq. Ž. 7, i follows h hrough pulse conrol he error re in qunum memory cn be reduced by fcor proporionl o he second order of he re of he pulse period o he noise correlion ime. In fc, by pplying more complice pulse sequences, he error re cn be furher reduced. For exmple, we my pply he pulse sequence I,s x,s z,s x, I,s x,s z, s x, I, PPP, wih he pulse period. In his sequence, conrol period consiss of eigh pulse periods. We cn similrly clcule he error re of he sysem conrolled by his pulse sequence. The resul is h he error re is reduced by fcor proporionl o Ž r. 4 D c. In generl, if we pply pulse sequence wih he conrol period consising of 2 nq pulse periods, he error re is ble o be reduced by fcor proporionl o Ž r. 2 n D c. In he bove, we considered decoherence of single qubi. Now, suppose h here re L qubis, described respecively by Puli s operors s. In l he inercion picure, he Hmilonin describing decoherence of L qubis cn be generlly expressed

4 42 ( ) L.-M. Dun, G.-C. GuorPhysics Leers A s Ž. Ž. Ž. HL s sl Gl, 8 l sx, y, z where he noise erms sisfy he condiions ² G Ž. : s0 nd l env y b b G Ž. G Ž. s f. Ž 9. ² : l l env ll ž c The bove descripion of mny qubi decoherence is quie generl. I includes independen decoherence nd cooperive decoherence s is specil cse w23 x. For reducing decoherence of his sysem, he conrol mehod is very simple. We need only pply he bove pulse sequence seprely on ech qubi. I is esy o know h he error re of he conrolled sysem will be reduced by fcor proporionl o Ž r. 2 n D c, where n depends on which pulse sequence is chosen. To mke ful-olern qunum compuion possible, noise in qunum ges should be suppressed s well s noise in qunum memory. The pulse conrol scheme cn be exended o include qunum ge operions. Any qunum ge cn be decomposed ino series of qunum conrolled-not Ž CNOT. ges ogeher wih some single-qubi row24 x. In relisic qunum compuer models ions w9,,4 x, he qunum CNOT ge is no described by single Hmilonin. I is composed of severl seps. The crucil seps re wo-qubi direc or indirec inercion processes, which re usully described by he following Heisenberg-like effecive Hmilonin w,4 x: H s g Ž. s Ps, Ž 0. g ll l l ll where g ll Ž., possibly dependen of ime, re coupling coefficiens. The Heisenberg-like inercion Ž 0., ogeher wih some single-qubi roions, suffice o consruc qunum CNOT. The ime required o perform he Heisenberg-like inercion is normlly much lrger hn he running ime of sinw,4 x. Hence, ny wo-bi or gle-qubi roions muli-bi qunum ge cn be decomposed ino series of slow Heisenberg-like inercions ogeher wih some fs single-qubi roions. The pulse conrol scheme is no used o reduce he error re of single-qubi roions. I is no prcicl o do so, since ech pulse Ž specil single-qubi operion. will inroduce n error re comprble wih h of single-qubi roions. However, he noise in he qunum ge minly comes from he slow Heisenberg-like inercion, nd we re mking progress if he environmenl noise in he Heisenberg-like inercion is suppressed by he scheme. This gol cn be esily ined. We need only synchronize he pulses cing on differen qubis, i.e., we pply he sme bi-flipping or phse-flipping pulses simulneously on ll of he qubis. The synchronized pulses induce eiher collecive bi flip mls l x or collecive phse flip mls l z, where ml denoes ensor produc. The Hmilonin Ž 0. remins unchnged under hese operions, i.e., Ž mls l x. HgŽ mls l x. sž mls l z. HgŽ mls l z. sh g. Ž. Hence, he Heisenberg-like inercion is no influenced by he pulses, wheres he environmenl noise in his process Ždescribed by he Hmilonin Ž 8.. is grely suppressed. There is remining quesion. Besides he Heisenberg-like inercions, ddiionl single-qubi roions re inroduced o perform qunum compuion. We need o show h hese roions cn be performed in wy h hey do no influence he noise-suppression pulse sequence. I is convenien o ke he pulse sequence I,s x,s z,s x,s z, PPP used in Eq. Ž. 4 s n exmple o illusre he generl resul. Firs, we should noe h he durion of single-qubi roions is comprble wih he pulse widh w nd hus much smller hn he pulse period. Hence, he roion is compleed in one of he pulse periods. To perform single-qubi roion R in he Ž nq. h conrol period Žsee Eq. Ž 4.., we my pply n ddiionl shor pulse R during he firs pule period, or pply shor pulse s x Rs x during he second pulse period, or pply s y Rs y Ž z s R s z. during he hird Ž fourh. pulse period. The effecive evoluion operor Ž. 4 in he Ž n q. h conrol period is hen replced by he following form: U ½ 4Ž nq. nq 4n 5 sx, y, z syt R exp yi H s G Ž. d. Ž 2.

5 ( ) L.-M. Dun, G.-C. GuorPhysics Leers A Obviously, single-qubi roion R is ined while he noise suppression is no influenced Žhe noise erms G re sill replced by he effecive. noise erms G. I is srighforwrd o exend his resul o include ny noise suppression pulse sequence. Hence, single-qubi roions cn be ined while suppressing noise in qunum memory nd he Heisenberg-like inercions by he pulse conrol mehod. This shows h our scheme is pplicble for reducing decoherence boh in qunum memory n in qunum compuion. We hve shown h by moduling he sysem hrough sequence of pulses much fser hn he decoherence process nd he Heisenberg-like inercions, bu sill much slower hn he single-qubi roions nd he free evoluion, he environmenl noise in qunum memory nd in qunum compuion cn be reduced while he oher dynmics remin unchnged. In he bove nlyses, we mke n idel ssumpion h he pulses inroduce no ddiionl noise. If he inccurcy of he pulse is considered, n ddiionl error re p0 will be inroduced by ech pulse Ž p0 is comprble wih he error re of single-qubi roions.. Now we esime in his circumsnce o wh moun he error re per qunum operion cn be reduced hrough pulse conrol Žhe qunum operion here refers o he sorge of qunum informion or he wo-bi or muli-bi qunum ges.. Le g nd pg denoe he running ime nd he environmen-induced error re of he operion, respecively, hen grd defines he num- ber of pulses pplied during he operion. Afer pulse conrol, he error re pproximely becomes ž 2 n D g p,p q p g g 0 c D g c p0g 2 nq GpgŽ 2nq., Ž 3 ž. 2np where he minimum is ined when ž 2 n p0gc 2 nq D s, 2np g which is he opiml vlue for he pulse period. The Ž. prmeer n in Eq. 3 depends on which pulse 2 n sequence is chosen. If 2 n is considerbly lrge, p rp, Ž p. rž p. g g 0 g g c. For environmen-induced error, p cn be pproximed by p, r w2 x g g g dec, where dec is he decoherence ime. Wih his pprox- imion, we hve p grp g,p0decr c. The error re is reduced by very smll fcor p0decr c. This suggess h he pulse conrol mehod is very effecive. Compred wih oher noise suppression schemes, he pulse conrol mehod hs severl remrkble feures. Firs, i coss no ddiionl qubis. This is n imporn feure since wih he presen echnology, qunum compuing resources re sill very sringen w0,,25 x. Only smll qunum sysems hve been demonsred experimenlly. For hese sysems, i is impossible o perform ful-olern qunum error correcion, bu he pulse conrol scheme works well. Second, in he pulse conrol scheme, no encoding, decoding, nd error correcion re required, nd no mesuremen is performed. Hence, in conrs o qunum error correcion, his scheme inroduces no slowdown of he compuion speed. Ls bu no he les, pulse conrol is relw9, x. The pulse conrol mehod, combined wih ful- ively mure echnology in experimens olern qunum error correcion, my ulimely mke relible qunum compuion possible. Acknowledgemens This projec ws suppored by he Nionl Nurl Science Foundion of Chin. References wx P.W. Shor, in Proc. of he 35h Annul Symposium on Foundions of Compuer Science ŽIEEE Press, Los Almios, CA, 994., pp wx 2 S. Lloyd, Science 273 Ž wx 3 L.K. Grover, Phys. Rev. Le. 79 Ž wx 4 P.W. Shor, Phys. Rev. A 52 Ž 995. R2493. wx 5 A.M. Sene, Phys. Rev. Le. 77 Ž wx 6 D.P. DiVincenzo, P.W. Shor, Phys. Rev. Le. 77 Ž wx 7 E. Knill, R. Lflmme, W.H. Zurek, Science 279 Ž wx 8 J. Preskill, Proc. R. Soc. London A 454 Ž wx 9 J.I. Circ, P. Zoller, Phys. Rev. Le. 74 Ž

6 44 ( ) L.-M. Dun, G.-C. GuorPhysics Leers A w0x C. Monroe e l., ibid 75 Ž wx N.A. Gershenfeld, I.L. Chung, Science 275 Ž w2x I.L. Chung, Nure 393 Ž w3x D.G. Cory e l., LANL e-prin qun-phr w4x B.E. Kne, Nure 393 Ž ; D.P. DiVincenzo, ibid 393 Ž w5x C.P. Slicher, Principles of Mgneic Resonnce, 3rd. ed. Springer-Verlg, New York, 990. w6x L. Viol, S. Lloyd, Phys. Rev. A 58 Ž w7x L.M. Dun, G.C. Guo, LANL e-prin qun-phr w8x D. Vili, P. Tombesi, Phys. Rev. A 59 Ž w9x L. Viol, E. Knill, S. Lloyd, Phys. Rev. Le. 82 Ž w20x P. Znrdi, Phys. Le. A 258 Ž w2x D.P. DiVincenzo, Science 270 Ž w22x B. Schumcher, Phys. Rev. A 54 Ž w23x L.M. Dun, G.C. Guo Phys. Rev. A 56 Ž ; Phys. Rev. Le. 79 Ž w24x S. Lloyd, ibid 75 Ž w25x W.S. Wrren, N.A. Gershenfeld, I.L. Chung, Science 277 Ž