John Preskill, Caltech KITP 7 June 2003

Transcription

1 Topologil quntum omputing for eginners John Preskill, Clteh KITP 7 June

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4 Kitev Freedmn

5 Kitev Freedmn Kitev, Fult-tolernt quntum omputtion y nyons (997). Preskill nd Ogurn, Topologil quntum omputtion (997). Preskill, Fult-tolernt quntum omputtion (997). Freedmn, Lrsen, nd Wng, A modulr funtor whih is universl for quntum omputtion (2000). Freedmn, Kitev, nd Wng, Simultion of topologil field theories y quntum omputers (2000). Mohon, Anyons from non-solvle groups re suffiient for universl quntum omputtion (2003). Mohon, Anyon omputers with smller groups (2004).

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7 Astrt for "Topologil quntum omputing for eginners," y John Preskill I will desrie the priniples of fult-tolernt quntum omputing, nd explin why topologil pprohes to fult tolerne seem espeilly promising. A two-dimensionl medium tht supports elin nyons hs topologil degenery tht n exploited for roust storge of quntum informtion. A system of n nonelin nyons in twodimensions hs n exponentilly lrge topologilly proteted Hilert spe, nd quntum informtion n e proessed y riding the nyons. I will disuss in detil two ses where nonelin nyons n simulte quntum iruit effiiently: fluxons in "nonelin superondutor," nd "Fioni nyons" with espeilly simple fusion rules.

8 Quntum Computtion Feynmn 8 Deutsh 85 Shor 94

9 A omputer tht opertes on quntum sttes n perform tsks tht re eyond the pility of ny oneivle lssil omputer. Feynmn 8 Deutsh 85 Shor 94

10 Finding Prime Ftors =??

11 Shor 94 Finding Prime Ftors =

12 Quntum omputer: the model () Hilert spe of n quits: spnned y ( 2 n ) H = C n n 2 0 Importnt: the Hilert spe is equipped with nturl tensorprodut deomposition into susystems. n C = C C C C times Physilly, this deomposition rises from sptil lolity. Elementry opertions ( quntum gtes ) tht t on smll numer of quits (independent of n) re esy; opertions tht t on mny quits (inresing with n) re hrd. n 2 { } x = x x x x, x 0, n (2) Initil stte: = 0 n

13 Quntum omputer: the model (3) A finite set of fundmentl quntum gtes: { U U U U } 2 3 n,,, G Eh gte is unitry trnsformtion ting on ounded numer of quits. The gtes form universl set: ritrry unitry trnsformtions n e onstruted, to ny speified ury, s quntum iruit onstruted from the gtes: U = (Universl gtes re generi.) Importnt: One universl set of gtes n simulte nother effiiently, so there is notion of omplexity tht is independent of the detils of the quntum hrdwre.

14 Quntum omputer: the model (4) Clssil ontrol: The onstrution of quntum iruit is direted y lssil omputer, i.e., Turing mhine. (We re not interested in wht quntum iruit n do unless the iruit n e designed effiiently y lssil mhine.) (5) Redout: At the end of the quntum omputtion, we red out the result y mesuring, i.e., projeting onto the sis σ { 0, } z (We don t wnt to hide omputtionl power in the ility to perform diffiult mesurements.)

15 Quntum omputer: the model Clerly, the model n e () n quits (2) initil stte (3) quntum gtes (4) lssil ontrol (5) redout simulted y lssil omputer with ess to rndom numer genertor. But there is n exponentil slowdown, sine the simultion involves mtries of exponentil size. The quntum omputer might solve effiiently some prolems tht n t e solved effiiently y lssil omputer. ( Effiiently mens tht the numer of quntum gtes = polynomil of the numer of its of input to the prolem.)

16 Quntum Error Corretion Shor 95 Stene 95

17 Quntum informtion n e proteted, nd proessed fult-tolerntly. Shor 95 Stene 95

18 Quntum Computer Deoherene Environment ERROR! If quntum informtion is leverly enoded, it n e proteted from deoherene nd other potentil soures of error. Intrite quntum systems n e urtely ontrolled.

19 Two Physil Systems Wht is the differene etween: A: Humn B: Chip Imperfet hrdwre. Hierrhil rhiteture with error orretion t ll sles... Relile hrdwre. Informtion proessing prevents informtion loss.

20 Topology Quntum Gte Noisy Gte

21 Topologil quntum omputtion (Kitev 97, FLW 00) nnihilte pirs? rid rid Kitev rid time rete pirs Freedmn

22 Topologil quntum omputtion (Kitev 97, FLW 00) time Physil fult tolerne with nonelin nyons: unontrolled exhnge of quntum numers will e rre if prtiles re widely seprted, nd therml nyons re suppressed...

23 Models of (nonelin) nyons A model of nyons is theory of two-dimensionl medium with mss gp, where the prtiles rry lolly onserved hrges. We define the model y speifying:. A finite list of prtile lels {,,, }. These indite the possile vlues of the onserved hrge tht prtile n rry. If prtile is kept isolted from other prtiles, its lel never hnges. There is speil lel 0 inditing trivil hrge, nd hrge onjugtion opertor C: (where 0=0). (Note: for prtile you my red punture. ) 2. Rules for fusing (nd splitting). These speify the possile vlues of the hrge tht n result when two hrged prtiles re omined. 3. Rules for riding. These speify wht hppens when two neighoring prtiles re exhnged (or when one is rotted y 2π) n

24 Fusion rules: Fusion vetor spe: Fusion = = N 0 V V V dim( V ) N µ (µ =, 2, 3,, N ) µ

25 Fusion rules: Fusion vetor spe: Fusion = = N 0 V V V dim( V ) N 0 0 The hrge 0 fuses trivilly, nd is the unique lel tht n fuse with to yield hrge 0. 0

26 Fusion An nyon model is sid to e nonelin if for some, nd, dim( V ) N 2. Then there is topologil Hilert spe tht n enode nontrivil quntum informtion. This enoding is nonlol; the informtion is olletive property of the two nyons, not lolized on either prtile. When the prtiles with lels nd re fr prt, different sttes in the topologil Hilert spe look identil to lol oservers. In prtiulr, the quntum sttes re invulnerle to deoherene due to lol intertions with the environment. Tht is why we propose to use this enoding in quntum omputer. µ

27 Fusion µ When we hide the quntum stte from the environment, we hide it from ourselves s well! But, when we re redy to red out the quntum stte (for exmple, t the onlusion of quntum omputtion), we n mke the informtion lolly visile gin y ringing the two prtiles together, fusing them into single ojet. Then we sk, wht is this ojet s lel? In ft, it suffies (for universl quntum omputtion) to e le to distinguish the lel = 0 from 0. It is physilly resonle to suppose tht we n distinguish nnihiltion into the vuum ( = 0) from lump tht is unle to dey euse of its onserved hrge ( 0).

28 Anywy, with nonelin nyons we n exploit topology not just to store quntum informtion, ut lso to proess it! Aelin vs. nonelin Aelin nyon models n lso e used for roust quntum memory, e.g., model of 2 fluxons nd their dul 2 hrges. A quit is relized euse the 2 flux in hole n e either trivil or nontrivil (the informtion is rried y the lels themselves, not y the fusion sttes). This informtion is hidden from the environment y mking the holes lrge nd keeping them fr prt (to prevent flux from tunneling from one hole to nother, or to the outside edge, nd to prevent the world lines of hrges from winding out holes). -- Kitev (996) However, this informtion my not e hrder to red out. We d need to ontrt hole to see if prtile ppers, or perform delite interferene experiment to detet the flux, or Alterntively, y mixing the 2 with eletromgneti U(), we might do the redout vi Senthil-Fisher type experiment (i.e., one tht would tully work)! -- Ioffe et l. (2002)

29 Assoitivity of fusion: the F-mtrix ( ) = ( ) µ ν = e' µν ' ( d ) e ' µ ' ν F ' e e eµν ν µ d d There re two nturl wys to deompose the topologil d Hilert spe V of three nyons in terms of the fusion spes of pirs of prtiles. These two orthonorml ses re relted y unitry trnsformtion, the F-mtrix.

30 Briding: the R-mtrix R : V V : µ = µ ( ) R µ µ µ When two neighoring nyons re exhnged ounterlokwise, their totl hrge is unltered; sine the prtiles swp positions, the fusion spe V hnges to the isomorphi spe V. This isomorphism is represented y unitry mtrix, the R-mtrix. The R-mtrix lso determines the topologil spin of the lel, i.e., the phse quired when the prtile is rotted y 2π: 2πiJ 0 e = R

31 Models of (nonelin) nyons A model of nyons is theory of two-dimensionl medium with mss gp, where the prtiles rry lolly onserved hrges. We define the model y speifying:. A finite lel set {,,, }. 2. The fusion rules = N 3. The F-mtrix (expressing ssoitivity of fusion). 4. The R-mtrix (riding rules). These determine representtion of the mpping lss group (riding plus 2π rottions), nd define unitry topologil modulr funtor (UTMF), the two-dimensionl prt of (2+)- dimensionl topologil quntum field theory (TQFT) --- relted to (+)-dimensionl rtionl onforml field theory (RCFT) n

32 Exmple: Yng-Lee (Fioni) Model 0 or The hrge tkes two possile vlues: 0 (trivil) nd (nontrivil, nd self-onjugte). Anyons hve hrge.two nyons n fuse in either of two wys: = 0+ This is the simplest of ll nonelin nyon models. Yet its deeptively simple fusion rule hs profound onsequenes. In prtiulr, the fusion rule determines the F-mtrix nd R- mtrix uniquely; the resulting nontrivil riding properties re dequte for universl quntum omputtion (pointed out y Kupererg).

33 Nonelin Anyons: Yng-Lee model Suppose n nyons hve trivil totl hrge 0. Wht is the dimension of the Hilert spe? 0 or 0, 0, 0, 0, 0, 0, 0, The distinguishle sttes of n nyons ( sis for the Hilert spe) re leled y inry strings of length n-3. But it is impossile to hve two zeros in row: 0 0

34 Nonelin Anyons: Yng-Lee model Suppose n nyons hve trivil totl hrge 0. Wht is the dimension of the Hilert spe? 0 or 0, 0, 0, 0, 0, 0, 0, The distinguishle sttes of n nyons ( sis for the Hilert spe) re leled y inry strings of length n-3. But it is impossile to hve two zeros in row: Therefore, the dimension is Fioni numer: D = 2, 3, 5, 8, 3, 2, 34, 55, 89, Asymptotilly, the numer of quits enoded y eh nyon is: ( ) log2φ = log / 2 = log 2(.68) =.694

35 Nonelin Anyons: Yng-Lee model Asymptotilly, the numer of quits enoded y eh nyon is: ( ) log2φ = log / 2 = log 2(.68) =.694 We sy tht d = φ is the (quntum) dimension of the Fioni nyon 0 or This ounting vividly illustrtes tht the quits re nonlol property of the nyons, nd tht the topologil Hilert spe hs no prtiulrly nturl deomposition s tensor produt of smll susystems. Anyons hve some nonlol fetures, ut they re not so nonlol s to profoundly lter the omputtionl model (the riding of nyons n e effiiently simulted y quntum iruit)

36 The quntum dimension Every nyon lel hs quntum dimension, whih we my define s follows: Imgine reting two prtile-ntiprtile pirs, nd then fusing the prtile from one pir with the ntiprtile of the other =, Annihiltion ours with proility /d 2. This is nturl generliztion of the se where the hrge is n irreduile representtion R of group G, where the quntum dimension is just the dimension R of the representtion (whih ounts the numer of olors going round the loop). But there is no logil reson why dimension defined this wy must e n integer, nd in generl it isn t n integer. = d

37 The quntum dimension There is more onvenient normliztion onvention for prtilentiprtile pirs... Eh time we dd nother tooth to the sw, it ost us nother ftor of /d. We n ompenste for tht d d d ftor y weighting eh pir retion or nnihiltion even y ftor of d. d d d = d With this onvention, losed loop hs weight, s though we were ounting olors d Now we n deform the world line of prtile (e.g., dding nd removing teeth ) without ltering the vlue of digrm.

38 The quntum dimension d d = = =, µ µ µ =, µ µ µ = N = N d Therefore, the vetor of quntum dimensions is the (Perron- Froenius) eigenvetor of eh fusion rule mtrix, with eigenvlue d : N d d d N d d d ( ) = = ( )

39 The quntum dimension N = u d u +, u = d D N = dim( V ) = NN N N 2 n 2 6 { } i n ( ) n n dd = N = u d u + = 2 + D Thus the quntum dimension ontrols the rte of growth of the n-prtile Hilert spe. The normliztion ftor D= d is lled the totl quntum dimension of the nyon model. 2 n-2

40 The quntum dimension Wht if we rete pirs of different types, nd then fuse? ( dd) p ( ) = µ µ µ = µ µ µ = N = N d p ( ) = N d (generlizes wht we found for the se = ) dd

41 The quntum dimension p ( ) = N d dd Suppose we rete dense gs of nyons (y quenhing ), with n ritrry initil distriution of prtile types. Then we let the gs nnel, ut not ompletely. The distriution of prtile types onverges to stedy stte distriution stisfying:, ppp ( ) = p 2 2 The solution to this eqution is: p = d D prtile popultions proportionl to squre of quntum dimension.

42 Briding: the B-mtrix d d B: V V : For the n-nyon Hilert spe, we my use the stndrd sis: n- n n-2 The effet of riding n e expressed in this sis: d e = e µν ' ' ( d ) e ' µ ν B ' eµν e d And the mtrix B is determined y R nd F: F R F

43 Topologil quntum omputtion (Kitev 97, FLW 00) nnihilte pirs? rid rid Kitev rid time rete pirs Freedmn

44 Topologil quntum omputtion. Crete pirs of prtiles of speified types. 2. Exeute rid. 3. Fuse neighoring prtiles, nd oserve whether they nnihilte. Clim: This proess n e simulted effiiently y quntum iruit. Need to explin:. Enoding of topologil Hilert spe. 2. Simultion of riding (B-mtrix s two-qudit gte). 3. Simultion of fusion (F-mtrix plus one-qudit projetive mesurement) n-2 n-3 n- n Although the topologil vetor spes re not themselves tensor produts of susystems, they ll fit into tensor produt of d-dimensionl systems, where this qudit is the totl fusion spe of three nyons ( ) ( n 2) V 0,, H d d =,, N 0

45 Topologil quntum omputtion n-2 n- 2 3 n n-3 n ( ) ( 2) H d Therefore, the topologil model is no more powerful thn the quntum iruit model. But is it s powerful? The nswer depends on the model of nyons, nd in prtiulr on the properties of the R-mtrix nd F-mtrix. To simulte quntum iruit, we enode quits in the topologil vetor spe, nd use riding to relize set of universl quntum gtes ting on the quits. Tht is, the imge of our representtion of the rid group B n on n strnds should e dense in SU(2 r ), for some r liner in n. Exmple: in the Fioni model, we n enode quit in the twodimensionl Hilert spe of four nyons with trivil totl 0 hrge. V { 0,} But wht re R nd F in this model?

46 Consisteny of riding nd fusing The R-mtrix (riding), nd the F-mtrix (ssoitivity of fusing) re highly onstrined y lgeri onsisteny requirements (the Moore-Seierg polynomil equtions). In the se of the Fioni model, these equtions llow us to ompletely determine R nd F from the fusion rules. By sequene of F-moves nd R-moves, we otin n isomorphism etween two topologil Hilert spes, tht is, reltion etween two different nonil ses. This reltion must not depend on the prtiulr sequene of moves, only on the sis we strt with nd the sis we end up with. For exmple, there re 5 different wys (without ny exhnges) to fuse five prtiles, relted y F-moves: F d e Pentgon eqution: ( 5 ) ( 5 F ) 34 F2 ( ) ( 5 ) ( 5 F ) 23 F e4 F234 d = e d e

47 Consisteny of riding nd fusing F R R F R F F R Hexgon eqution: ( 4 ) 4 ( 4 ) ( 4 ) = F R F R F R Furthermore, if the pentgon nd hexgon equtions re stisfied, then ll sequenes of F- nd R-moves from n initil sis to finl sis yield the sme isomorphism! A systemti (in priniple) proedure for onstruting nyon models:. Assume fusion rule. 2. Solve pentgon nd hexgon equtions for R nd F. -- If no solutions, the fusion rules re inomptile with lol quntum physis. -- If multiple solutions, eh is vlid model.

48 Exmple: Fioni model = Σ F R : F 4 πi /5 τ τ e 0 =, R=, τ = ( 5 )/2= φ 2 πi /5 τ τ 0 e This solution is unique (side from freedom to redefine phses nd tke the prity onjugte). Furthermore, produts of the nonommuting mtries R nd FRF - (representing the genertors of the rid group B 3 ) re dense in SU(2).

49 Exmple: Fioni model We enode quit in four nyons. To simulte quntum iruit, we need to do (universl) two-quit gtes. The two-quits re emedded in the 3-dimensionl Hilert spe of eight nyons. The representtion of B 3 determined y our R nd F mtries is universl i.e., dense in SU(3), so in prtiulr we n pproximte ny SU(4) gte ritrrily well with some finite numer of exhnges. If we fix ury of the pproximtion to the gte, we n use quntum error- orreting odes nd fult-tolernt simultion to perform n effiient nd relile quntum omputtion. Here quntum-error orretion might e needed to orret for the (smll) flws in the gtes, ut not to orret for storge errors.

50 For exmple, we n use quntum teleporttion protool for lekge orretion (in effet, this turns quntum lekge into lssil lekge, whih is esier to detet nd orret). Lekge The omputtion tkes ple in the r-quit suspe of system of 4r nyons. As errors umulte, the stte of the omputer might drift our of this suspe (the lekge prolem). But we n inlude lekge orretor gtes in our simultion. This gte is the identity ting on dt in the omputtionl spe, ut reples leked quit y the stndrd stte 0 in the omputtionl spe. unleked dt leked dt Lekge Corretor Lekge Corretor unhnged unleked dt 0

51 Topologil quntum omputtion To summrize, we n simulte universl quntum omputer using (for exmple) Fioni nyons, if we hve these pilities:. We n rete pirs of prtiles. 2. We n guide the prtiles long speified rid. 3. We n fuse prtiles, nd distinguish omplete nnihiltion from inomplete nnihiltion. -- The temperture must e smll ompred to the energy gp, so tht stry nyons re unlikely to e exited thermlly. -- The nyons must e kept fr prt from one nother ompred to the orreltion length, to suppress hrge-exhnging virtul proesses, exept during the initil pir retion nd the finl pir nnihiltion.

52 There would still e more to do, though For exmple, this would e lssifition of gpped two-dimensionl ulk theories, nd one ulk theory n orrespond to more thn one (+)-dimensionl theory desriing edge exittions. And of ourse, we would like to know, oth for prtil nd theoretil resons, whether the model n e relized roustly with some lol Hmiltonin (nd how to relize it). (Nonelin) nyons An nyon model is hrterized y its lel set, fusion rules, F-mtrix, nd R-mtrix. Clssifying the models (finding ll solutions to the pentgon nd hexgon equtions) is n importnt (hrd) unsolved mthemtil prolem. We know how to find some exmples (e.g., Chern-Simons theories), ut we don t know how rih the possiilities re. Suh lssifition would e n importnt step towrd lssifying topologil order in two dimensions. µ F R

53 Topologil quntum memory Kitev 96 Quits n reside in holes in plnr rry, where the holes rry Z 2 hrge or flux. Then the quntum memory is topologilly stle, ut nontopologil ouplings etween holes re needed to omplete set of universl gtes. This sheme might e relizle in suitly designed Josephson-juntion rrys, whih hve phse tht n e interpreted s ondenste of ojets with hrge 4e. A hole in the rry n rry hrge 2e or flux Φ 0 /2=2π/4e. Ioffe et l. 02

54 Quntum mny-ody physis: Exoti phses in optil ltties Atoms n e trpped in n optil lttie. The lttie geometry nd intertions etween neighors n e hosen y the mteril designer (diretion-dependent nd spin dependent tunneling etween sites). In prtiulr, Dun, Lukin, nd Demler (ond-mt/020564) hve desried how Kitev s honeyom lttie model, whih supports nonelin nyons, n e simulted using n optil lttie.

55 Topologil quntum omputing for eginners John Preskill, Clteh KITP 7 June