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

Transcription

1 Spin Networks n Anyoni Topologil Quntum Computing L. H. Kuffmn, UC 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 llinois t Chigo, Chigo, llinois , USA Deprtment of Computer Siene n Eletril Engineering, University of Mryln Bltimore County, 1000 Hilltop Cirle, Bltimore, MD 21250, USA

2 Originl Motivtion: Use topologil strutures to solve prolem of eoherene in physil reliztion of quntum omputing systems. The Hope: physil system suh s quntum Hll flui, hving topologil fetures, will e resistnt to perturtion n so give rise to relile n slle omputing..

3 Quntum Hll systems re ttrtive: They re relte to Conforml Fiel Theory n Chern-Simons-Witten Theory on physil sie n to funmentl low imensionl topology. Gtes woul e onstrute vi the riing of olletive eletroni exittions in two imensionl mei. nvestigtion of the topology les to rih unitry representtions of the Artin ri group.

4 Representtions of the ri group n generte ll unitry trnsformtions neee for quntum omputing. Unitry representtions of the Artin ri group hve s muh import s the symmetri groups for the struture of unitry groups.

5 Unitry representtions of the Artin ri group give rise to quntum lgorithms for omputing the originl Jones polynomil, the olore Jones polynomil n the Witten-Reshetikhin-Turev invrints of three-mnifols.

6 Topology n Quntum Computing: Reltionships etween topology n quntum omputing re signifint for 1. Future physis n physil pplitions. 2. The mthemtis of quntum informtion. 3. The struture of quntum lgorithms.

7 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.

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

9 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 ust 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. t 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 f0i; 1ig: H D p ; S D i ; T D e i4

10 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. n [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.

11 Briing n the Yng-Bxter Eqution R R R R R R R R (R )( R)(R ) ( R)(R )( R).

12 Representtive Exmples of Unitry Solutions to the Yng-Bxter Eqution tht re Universl Gtes. R 1/ / 2 0 1/ 2 1/ / 2 1/ 2 0 1/ / 2 Bell Bsis Chnge Mtrix R R R Swp Gte with Phse te.

13 Quntum Hll Effet

14 Briing Anyons Λ Reoupling Proess Spes

15 Non-Lol Briing is nue vi Reoupling F R F B F RF

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

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

18 Knots n Links s s 1 2 Bri Genertors Hopf Link s s 3 1 s s Trefoil Knot s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 3 s s 3 1 Figure Eight Knot

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

20 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 llinois t Chigo, 851 S. Morgn Street, Chigo, L , 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 ) t 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 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 1x Step #2: pplition of U on the NMR prout opertor 1x : 1 0 U 1 ( 0 U ) ( ) ( 0 U ) 1 ( ) 0 U 2 U 0 Step #3: mesurement of 1x n 1y : 0 U tr { 1x ( U 0 )} 1 1 ( tr{ U}) U tr { 1y ( U 0 ) } 1 1 ( tr{ U}) 2 2 U, U nste 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 n mesure 1x n 1y in orer to otin 1x the tre of the originl mtrix U. nepenent 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 1x n 1y n e omplishe in one single-sn experiment. NMR pulse sequene S U 1 mens S - z - z J S U 1 U 1 U 1 S U 1 mens S y J S - y z z S U 2 mens U 1 U 2 U 1 U 2 S y - z - z J - y S S U 2 mens 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 1x, mesurement of 1y, referene for 1x, n referene for 1y. f 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 + A 0 The Jones Polynomils n e reonstrute out of the NMR experiments y: 3 -w( L) ( 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 ( 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,

21 q-deforme Spin Networks ~ 2-2 -A - A 2 1/δ A + A n 1 1 n 1 1 n... n n / n n+1 n n strns δ n+1 - n n {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 k 45 i + + k i + k

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

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

24 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

25 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 ).

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

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

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

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

30 Σ i k Σ δ k k MoTet[ ] i i i i i k k k i New Reoupling Formul

31 The Reoupling Mtrix is Rel Unitry t Roots of Unity. i Σ i i i, i i M[,,,] i i T

32 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.

33 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.

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