Spin Networks and Anyonic Topological Quantum Computing L. H. Kauffman, UIC.

Transcription

1 Spin Networks n Anyoni Topologil Quntum Computing L. H. Kuffmn, UIC qunt-ph/ n qunt-ph/ Spin Networks n Anyoni Topologil Computing Louis H. Kuffmn n Smuel J. Lomono Jr. Deprtment of Mthemtis, Sttistis n Computer Siene (m/ 249), 851 South Morgn Street, University of Illinois t Chigo, Chigo, Illinois , USA Deprtment of Computer Siene n Eletril Engineering, University of Mryln Bltimore County, 1000 Hilltop Cirle, Bltimore, MD 21250, USA

2 Journl of Knot Theory n Its Rmifitions Vol. 16, No. 3 (2007) Worl Sientifi Pulishing Compny q-deformed SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION LOUIS H. KAUFFMAN Deprtment of Mthemtis, Sttistis n Computer Siene (m/ 249), 851 South Morgn Street, University of Illinois t Chigo, Chigo, Illinois , USA kuffmn@ui.eu SAMUEL J. LOMONACO JR. Deprtment of Computer Siene n Eletril Engineering, University of Mryln Bltimore County, 1000 Hilltop Cirle, Bltimore, MD 21250, USA lomono@um.eu Aepte 10 July 2006 ABSTRACT We review the q-eforme spin network pproh to Topologil Quntum Fiel Theory n pply these methos to proue unitry representtions of the ri groups tht re ense in the unitry groups. Our methos re roote in the rket stte sum moel for the Jones polynomil. We give our results for lrge lss of representtions se on vlues for the rket polynomil tht re roots of unity. We mke seprte n self-ontine stuy of the quntum universl Fioni moel in this frmework. We pply our results to give quntum lgorithms for the omputtion of the olore Jones polynomils for knots n links, n the Witten Reshetikhin Turev invrint of three mnifols. Keywors: Knot; link; Reiemeister move; rket polynomil; Jones polynomil; olore Jones polynomils; Kuffmn polynomil; spin network; quntum omputtion; quntum omputer; Temperley Lie lger; reoupling theory; Fioni moel; ri group; unitry representtion. Mthemtis Sujet Clssifition 2000: 57M27

3 Quntum Mehnis in Nutshell 0. A stte of physil system orrespons to unit vetor S> in omplex vetor spe. U 1. (mesurement free) Physil proesses re moele y unitry trnsformtions pplie to the stte vetor: S> -----> U S> 2. If S> z1 e1> + z2 e2> zn en> in mesurement sis { e1>, e2>,..., en>}, then mesurement of S> yiels ei> with proility zi ^2.

4 Preprtion,Trnsformtion, Mesurement. Psi <T U S> Psi*Psi <S U* T> <T U S> U <T S> U*

5 Quit A quit is the quntum version of lssil it of informtion. 0> + 1> mesure 0> 1> pro ^2 pro ^2

6 0> 1> 0> 1> - 1> 0> 1> 0> 0> Mh-Zener Interferometer 0> 1> 1> 0> 0> - 1> [ ] H [ 0 1 ] /Sqrt(2) M 1 HMH [ ]

7 Quntum Gtes re unitry trnsformtions enliste for the purpose of omputtion. CNOT CNOT 00> 00> CNOT 01> 01> CNOT 10> 11> CNOT 11> 10>

8 Quntum Computtion of the Tre of Unitry Mtrix U 1. A goo exmple of quntum lgorithm. 2. Useful for the quntum omputtion of knot polynomils suh s the Jones polynomil.

9 Hmr Test 0> phi> H U H Mesure 0> 0> ours with proility 1/2 + Re[<phi U phi>]/2

10 Grover s Algorithm (1996) [O(Sqrt(N)) time, O(log N) storge spe] Given n unsorte tse with N entries. {0,1,2,...,N-1} Prolem: Fin prtiulr element w in the tse. Form N-imensionl stte spe V. H n oservle ting on V with N istint eigenvlues. { 0>, 1>, 2>,... N-1>} sis for V.

11 Introue stte vetor s> (1/Sqrt(N)) Sum x> sum is over the sis of V. w> s> Ie: Use unitry opertions to rotte s> into the w> iretion.

12 w> Pi/2 - Thet s> Thet U(w) refletion in plne perp to w>. U(s) refletion in s>. s > U(s)U(w) s> is rotte towr w> y 2 x Thet. Do this pprox Pi Sqrt(N)/4 times. For N lrge the proility of not oserving w> is O(1/N).

13 Polynomil-Time Algorithms for Prime Ftoriztion n Disrete Logrithms on Quntum Computer rxiv:qunt-ph/ v2 25 Jn 1996 Peter W. Shor Astrt A igitl omputer is generlly elieve to e n effiient universl omputing evie; tht is, it is elieve le to simulte ny physil omputing evie with n inrese in omputtion time y t most polynomil ftor. This my not e true when quntum mehnis is tken into onsiertion. This pper onsiers ftoring integers n fining isrete logrithms, two prolems whih re generlly thought to e hr on lssil omputer n whih hve een use s the sis of severl propose ryptosystems. Effiient rnomize lgorithms re given for these two prolems on hypothetil quntum omputer. These lgorithms tke numer of steps polynomil in the input size, e.g., the numer of igits of the integer to e ftore. Keywors: lgorithmi numer theory, prime ftoriztion, isrete logrithms, Churh s thesis, quntum omputers, fountions of quntum mehnis, spin systems, Fourier trnsforms AMS sujet lssifitions: 81P10, 11Y05, 68Q10, 03D10

14 Universl Gtes A two-quit gte G is unitry liner mpping G : V V V V where V is two omplex imensionl vetor spe. We sy tht the gte G is universl for quntum omputtion (or just universl) if G together with lol unitry trnsformtions (unitry trnsformtions from V to V ) genertes ll unitry trnsformtions of the omplex vetor spe of imension 2 n to itself. It is well-known [44] tht CNOT is universl gte. Lol Unitries re generte (up to ensity) y smll numer of gtes. Expliit gte reliztion in the sis fj0i; j1ig: H D p ; S D i ; T D e i4

15 A gte G is universl iff G is entngling. A gte G, s ove, is si to e entngling if there is vetor αβ α β V V suh tht G αβ is not eomposle s tensor prout of two quits. Uner these irumstnes, one sys tht G αβ is entngle. In [6], the Brylinskis give generl riterion of G to e universl. They prove tht two-quit gte G is universl if n only if it is entngling.

16 An Entngle Stte

17 An Entnglement Criterion Remrk. A two-quit pure stte φ is entngle extly when ( ) 0. It is esy to use this ft to hek when speifi mtrix is, or is not, entngling. The Bell Sttes R 00 (1/ 2) 00 (1/ 2) 11, R 01 (1/ 2) 01 + (1/ 2) 10, R 10 (1/ 2) 01 + (1/ 2) 10, R 11 (1/ 2) 00 + (1/ 2) 11.

18 Briing n the Yng-Bxter Eqution R I I R R I I R I R R I R I I R (R I)(I R)(R I) (I R)(R I)(I R).

19 Let V e two omplex imensionl vetor spe. Briing Opertors re Universl Quntum Gtes Universl gtes n e onstrute from ertin solutions to the Yng-Bxter Eqution R: V V V V (R I)(I R)(R I) (I R)(R I)(I R).

20 Representtive Exmples of Unitry Solutions to the Yng-Bxter Eqution tht re Universl Gtes. te. R R R 0 1/ / 2 0 1/ 2 1/ / 2 1/ 2 0 1/ / Swp Gte with Phse R Bell Bsis Chnge Mtrix R + R* Sqrt[2]I Corresponing Link Invrint is Speil Cse of Homfly Poly. (virtul rossing orrespons to swp gte.)

21 Issues 1. Giving Universl Gte tht is topologil gives PARTIAL topologil quntum omputing euse the U(2) lol opertions hve not een me topologil. 2. Nevertheless, Yng-Bxter gtes re interesting to onstrut n help to isuss Topologil Entnglement versus Quntum Entnglement.

22 Quntum Entnglement n Topologil Entnglement An exmple of Arvin [1] mkes the possiility of suh onnetion even more tntlizing. Arvin ompres the Borromen rings (see figure 2) n the GHZ stte ψ ( β 1 β 2 β 3 α 1 α 2 α 3 )/ 2. ( 000> - 111>)/Sqrt(2) Is the Arvin nlogy only superfiil?!

23 Do we nee Quntum Knots? K> + K > K: proility ^2 K :proility ^2 K K Oserving Quntum Knot

24

25 The Temperley-Lie Ctegory Ientity Ω > Θ < Θ Ω < > Ω Θ > < U U U φ > Θ > ψ > { Ω Θ } > < 1 P Ω < 1 QPQQ { Ω > < } Θ Q { 1 < } { > PQP Θ P Ω φ > ψ > Θ Ω φ > ψ > Θ Ω The Key to Teleporttion

26 Digrmmti Mtries, Knots n Teleporttion i N i N M i M M M Figure 5 - Mtrix Composition

27 Quntum Link Invrints

28 y mesuring mplitue n phse in referene setting.. r.. Untying Knots y NMR: first experimentl implementtion of quntum lgorithm for pproximting the Jones polynomil Rimun Mrx 1, Anres Spörl, Amr F. Fhmy, John M. Myers, Louis H. Kuffmn, Smuel J. Lomono, Jr., Thoms-Shulte-Herrüggen, n Steffen J. Glser 1 Deprtment of Chemistry, Tehnil University Munih, Lihtenergstr. 4, Grhing, Germny 2 Hrvr Meil Shool, 25 Shttuk Street, Boston, MA 02115, U.S.A. 3 Goron MKy Lortory, Hrvr University, 29 Oxfor Street, Cmrige, MA 02138, U.S.A. 4 University of Illinois t Chigo, 851 S. Morgn Street, Chigo, IL , U.S.A. 5 University of Mryln Bltimore County, 1000 Hilltop Cirle, Bltimore, MD 21250, U.S.A. romp of the quntum lgorithm exmple #1 Trefoil exmple #2 Figure-Eight exmple #3 Borromen rings A knot is efine s lose, non-self-interseting urve tht is emee in three imensions. exmple: onstrution of the Trefoil knot: knot or link mke knot fuse the free ens mke it look nie strt with rope en up with Trefoil trelose ri J. W. Alexner prove, tht ny knot n e represente s lose ri (polynomil time lgorithm) genertors of the 3 strn ri group: unitry mtrix i e U 1 0 U 3 1 U Trefoil 1 U U 2 Figure Eight 2 U1 0 i sin(4 ) i e e sin(2 ) 1 U U Borrom. R. U i sin(6 ) i i sin(6 )sin(2 ) e e e sin(4 ) sin(4 ) U 2 i sin(2 ) i sin(6 )sin(2 ) i e e e sin(4 ) sin(4 ) -1-1 It is well known in knot theory, how to otin the unitry mtrix representtion of ll genertors of given ri goup (see Temperley-Lie lger n pth moel representtion ). The unitry mtries U1 n U 2, orresponing to the genertors 1 n 2 of the 3 strn ri group re shown on the left, where the vrile is relte i to the vrile A of the Jones polynomil y: A e. 1 2 The unitry mtrix representtions of n -1-1 re given y U1 n U 2. The knot or link tht ws expresse s prout of ri group genertors n therefore lso e expresse s prout of the orresponing unitry mtries. ontrolle unitry mtrix Step #1: from the 2x2 mtrix U to the 4x4 mtrix U: 1 0 U ( 0 U ) U I 1x Step #2: pplition of U on the NMR prout opertor I1x : 1 0 U 1 ( 0 U ) ( ) ( 0 U ) 1 ( ) 0 U 2 U 0 Step #3: mesurement of I 1x n I 1y : 0 U tr { I 1x ( U 0 )} 1 1 ( tr{ U}) U tr { I 1y ( U 0 ) } 1 1 ( tr{ U}) 2 2 U, U Inste of pplying the unitry mtrix we pply it s ontrolle vrint. This mtrix is espeilly suite for NMR quntum omputers [4] n other therml stte expettion vlue quntum omputers: you only hve to pply U to the NMR prout opertor I n mesure I1x n I1y in orer to otin 1x the tre of the originl mtrix U. Inepenent of the imension of mtrix U you only nee ONE extr quit for the implementtion of U s ompre to the implementtion of U itself. The mesurement of I1x n I1y n e omplishe in one single-sn experiment. NMR pulse sequene I S U 1 mens I S - z - z J I S U 1 U 1 U 1 I S U -1 1 mens I S y J I S - y z z I S U 2 mens -1-1 U 1 U 2 U 1 U 2 I S y - z - z J - y I S I S U -1 2 mens I S U 1 U 2 U 1 U 2 U 1 U 2 y J - y z z - y All knots n links n e expresse s prout of ri group genertors (see ove). Hene the orresponing NMR pulse sequene n lso e expresse s sequene of NMR pulse sequene loks, where eh lok orrespons to the ontrolle unitry mtrix U of one ri group genertor.. This moulr pproh llows for n esy optimiztion of the NMR pulse sequenes: only smll n limite numer of pulse sequene loks hve to e optimize.. NMR experiment Comprison of experimentl results, theoretil preitions, n simulte experiments, where relisiti inperfetions like relxtion, B1 fiel inhomogeneity, n finite length of the pulses re inlue. For eh t point, four single-sn NMR experiments hve een performe: mesurement of I1x, mesurement of I1y, referene for I1x, n referene for I1y. If neessry eh t point n lso e otine in one single-sn experiment Jones polynomil Jones Polynomil Trefoil": ( A + A - A ) 2-2 (- A - A ) Jones Polynomil Figure-Eight": Jones Polynomil Borromen rings": + 4A A - A - A + 3A - 2A + A - A -4 - A + 3A 0-2A -1 + A 0 The Jones Polynomils n e reonstrute out of the NMR experiments y: 3 -w( L) I( L) V (A)(- A ) ( tr{ U} + A [(-A -A ) -2]) L where: w( L) is the writhe of the knot or link L tr{ U} is etermine y the NMR experiments I( L) is the sum of exponents in the ri wor orresponing to the knot or link L Referenes: 1) 1) L. Kuffmn, AMS Contemp. Mth. Series, 305, eite y S. J. Lomono, (2002), (mth.qa/ ) 2) R. Mrx, A. Spörl, A. F. Fhmy, J. M. Myers, L. H. Kuffmn, S. J. Lomono, Jr., T. Shulte-Herrüggen, n S. J. Glser: pper in preprtion 3) Vughn F. R. Jones, Bull. Amer. Mth. So., (1985), no. 1, ) J. M. Myers, A. F. Fhmy, S. J. Glser, R. Mrx, Phys. Rev. A, (2001), 63, (qunt-ph/ ) 5) D. Ahronov, V. Jones, Z. Lnu, Proeeings of the STOC 2006, (2006), (qunt-ph/ ) 6) M. H. Freemn, A. Kitev, Z. Wng, Commun. Mth. Phys., (2002), 227,

29 SU(2) Representtions of the Artin Bri Group Theorem. If g + u n h + v re pure unit quternions,then, without loss of generlity, the ri reltion ghg hgh is true if n only if h + v, n φ g (v) φ h 1(u). Furthermore, given tht g + u n h + v, the onition φ g (v) φ h 1(u) is stisfie if n only if u v when u v. If u v then then g h n the ri reltion is trivilly stisfie. g + u h + v u v (^2 - ^2)/2^2

30 An Exmple. Let where os(θ) n sin(θ). Let g e iθ + i h + [( 2 s 2 )i + 2sk] where 2 +s 2 1 n 2 s Then we n reexpress g n h in mtrix 2 2 form s the mtries G n H. Inste of writing the expliit form of H, we write H F GF where F is n element of SU(2) s shown elow. G F ( e iθ 0 0 e iθ ( i is is i ) ) ing where one genert

31 SU(2) Fioni Moel τ 2 + τ 1. g e 7πi/10 f iτ + k τ h frf 1 fgf -1 {g,h} represents 3-strn ris, generting ense suset of SU(2).

32 We shll see tht the representtion lele SU(2) Fioni Moel in the lst slie extens eyon SU(2) to representtions of mny-strne ri groups rih enough to generte quntum omputtion.

33 Quntum Hll Effet

34 The qusi-prtile theory is onnete with Chern-Simons Theory n it explins the FQHE on the sis of nyons : prtiles tht hve non-trivil (not +1 or -1) phse hnge when they exhnge ples in the plne.

35 Nuler Physis B360 (1991) North-Holln NONABELIONS IN THE FRACTIONAL QUANTUM HALL EFFECT Gregory MOORE Deprmzent of Physis, Yle Uniersity, New Hen, CT 06511, USA Nihols READ Deprtments of Applie Physis n Physis, Yle Unirersity, New Ht'en, CT 06520, USA Reeive 31 My 1990 (Revise 5 Deemer 1990) Applitions of onforml fiel theory to the theory of frtionl quntum Hll systems re isusse. In prtiulr, Lughlin's wve funtion n its ousins re interprete s onforml loks in ertin rtionl onforml fiel theories. Using this point of view hmiitonin is onstrute for eletrons for whih the groun stte is known extly n whose qusihole exittions hve nonelin sttistis; we term these ojets "nonelions". It is rgue tht universlity lsses of frtionl quntum Hll systems n e hrterize y the quntum numers n sttistis of their exittions. The reltion etween the orer prmeter in the frtionl quntum Hll effet n the hirl lger in rtionl onforml fiel theory is stresse, n new orer prmeters for severl sttes re given.

36 3. Eletron wve funtions s onforml loks: Lughlin sttes n the hierrhy Let us return to the Lughlin stte in the is geometry:, 2] ~l~.gi, li,,(zl,---, zn) r l ( z, - zs) exp[ - ~ Y:lz, I, i < j (3.1) where q is n o integer [3]. In the thermoynmi limit this stte IOL; N ) esries flui groun stte with uniform numer ensity P0 - v/2z: 1/2zrq insie rius of orer 2~-N. The GL esription of this limit for normlize flui stte [t~ ) of slowly vrying ensity involves guge fiel i ~ ( z ) ~ f z - z ' z' (3.2) In the GL esription [4] this guge fiel ouples to the orer prmeter (whih hs hrge q; we set the hrge of the eletron to 1 from now on) n lso enters with Chern-Simons term q 4rr f z C ~ (3.3) in the tion. If we re intereste primrily in sttistis of exittions we my expet suh topologil terms in the tion to ply ominnt role - sine they ominte ll other terms t long istnes n low energies. On the other hn, it is now well known tht CSW theory (i.e. (2 + 1)-imensionl guge theory with only CS term in the tion) for n elin guge fiel is losely onnete to the (1 + 1)-imensionl onforml fiel theory known s the "rtionl torus" [1,5].

37 Briing Anyons Λ Reoupling Proess Spes

38 Proess Vetor Spes n Reoupling ε V( ) e f ε V( )

39 A B A B C C

40 Topologil Quntum Fiel Theory Trinion Proess Spes on Surfes Le to Three-Mnifol Invrints.

41 Non-Lol Briing is Inue vi Reoupling F R -1 F B F -1 RF

42 Proess Spes Cn e Aitrrily Lrge. With oherent reoupling theory, ll trnsformtions re in the representtion of one ri group.

43 Mthemtil Moels for Reoupling Theory with Briing ome from Comintion of Penrose Spin Networks n Knot Theory. See Temperley Lie Reoupling Theory n Invrints of Three-Mnifols y L. Kuffmn n S. Lins, PUP, 1994.

44 Brket Polynomil Moel for Jones Polynomil -1 A A A -1 A A -1 A -1 < > A < > + A < > -1 < > A < > + A < > < K > S < K S > δ S 1.

45 q-deforme Spin Networks ~ 2-2 -A - A 2 1/δ A + -1 A n 1 1 n 1 1 n... n n / n n+1 n n strns δ n+1 - n n-1 {n}! Σ σ ε S n -4 t( σ) (A ) 0 n -3 t( σ ) (1/{n}!) (A ) Σ σ ε S n ~ σ i j k 45 i + j j + k i + k

46 Projetors re Sums over Permutions, Lifte to Bris n Expne vi the Brket into the Temperley Lie Alger

47 Briing, Nturlity, Reoupling, Pentgon n Hexgon -- Automti Consequenes of the Constution R F F R F F F F F F R R F

48 P P P P Fioni Moel A e 3πi/5. ) ( 1/δ * P P P P 0 P 111 im(v ) 1 0 Forien * P P P P P P P P * P 1111 im(v ) 2 0 * * 0 > 1 > P P Temperley Lie Representtion of Fioni Moel

49

50 Fioni Moel

51 The Simple, yet Quntum Universl, Struture of the Fioni Moel A e 3πi/5. ) ( ( ) ( ) δ A 2 A 2 δ (1 + 5)/2, ( 1/ 1/ F 1/ 1/ ( ) A 4 0 R 0 A 8 ) ( ) τ τ τ τ ( e 4πi/5 0 0 e 2πi/5 exmple of ri group representtion ).

52 Spin Network Gymnstis

53 1/δ (δ 1/δ) (δ 1/δ) (δ 1/δ) δ 2 δ 1 Θ 1/δ Θ 2 (δ 1/δ) δ /δ

54 + + 1/ 2 Θ Θ / /Θ 2 Θ/ Τ /Θ 2

55 Θ(,, ) Θ(,, ) δ { } i j Σ j i j k [ ] Tet i k i Closure, Bule n Reoupling

56 The 6-j Coeffiients i k Σ j i { j } j k Σ j i { j } Θ(,, j ) Θ(,, j ) j j j k δ j { } i Θ(,, k) Θ(,, k ) k k i { } k Tet i [ ] k k Θ(,, k) Θ(,, k)

57 Lol Briing λ λ (+-)/2 (-1) A ('+'-')/2 x' x(x+2)

58 Θ(,, ) Reefining the Vertex is the key to otining Unitry Reoupling Trnsformtions. Θ(,, ) Θ(,, )

59 Σ j i k Σ j δ j k k MoTet[ ] i i j i j i i j j j k k k j j i j j j j j j New Reoupling Formul

60 The Reoupling Mtrix is Rel Unitry t Roots of Unity. i Σ j i j j i j i j, i j j i M[,,,] i j i j T -1

61 Theorem. Unitry Representtions of the Bri Group ome from Temperley Lie Reoupling Theory t roots of unity. A e iπ/2r Suffiient to Proue Enough Unitry Trnsformtions for Quntum Computing.

62 Quntum Computtion of Colore Jones Polynomils n WRT invrints. B P(B) Σ B(x,y) x y x, y Σ B(x,y) if 0 0 Σ x, y B(x,y) x 0 y 0 0 B(0,0) if 0 B(0,0) ( ) 2 B(0,0) ( ) 2 Nee to ompute igonl element of unitry trnsformtion. Use the Hmr Test.

63 Colore Jones Polynomil for n 2 is Speiliztion of the Durovnik version of Kuffmn polynomil. 4-4 A + A + δ -4 4 A + A + δ ( A - A ) - ( ) ( A - A ) - ( ) 8 A

64 Will these moels tully e use for quntum omputtion? Will quntum omputtion tully hppen? Will topology ply key role? Time will tell.