Measurement-Only Topological Quantum Computation

Transcription

1 Mesurement-Only Topologicl Quntum Computtion Prs Bonderson Microsoft Sttion Q University of Virgini Condensed Mtter Seminr October, 8 work done in collbortion with: Mike Freedmn nd Chetn Nyk rxiv:8.79 (PRL 8) nd rxiv:88.933

2 Introduction Non-Abelin nyons re believed to exist in certin gpped two dimensionl systems: - Frctionl Quntum Hll Effect (n5/, /5,?) - ruthentes, topologicl insultors, rpidly rotting bose condenstes, quntum loop gses/string nets? If they exist, they could hve ppliction in quntum computtion, providing nturlly ( topologiclly protected ) fult-tolernt hrdwre. Assuming we hve them t our disposl, wht opertions re necessry to implement topologicl quntum computtion?

3 Prticle Exchnge Sttistics y R x U[R] t y x

4 Prticle Exchnge Sttistics 3 (nd higher) sptil dimensions: R R nd R Only initil nd finl positions re topologiclly distinguished Sttistics chrcterized by permuttion group S n Bosons nd Fermions

5 Prticle Exchnge Sttistics sptil dimensions: R R Worldlines form topologiclly distinct brid configurtions Sttistics chrcterized by brid group B n

6 (n strnd) brid group B n R i R i

7 Yng - Bxter constrint : R R i R R R R i i i i i R i R i R i R i R i R i

8 Prticle Exchnge Sttistics sptil dimensions: R R Worldlines form topologiclly distinct brid configurtions Sttistics chrcterized by brid group B n This gives

9 Briding Sttistics One dim unitry reps of B n ssign phse to ech brid genertor: U [ R i ] e i Abelin nyons (bosons :, fermions : ) Higher dim reps of B n men Hilbert spce is multi-dimensionl, nd unitry mtrices re ssigned to brid genertors: U[ R i ] U non-abelin nyons!

10 Toy model of Abelin Anyons: chrge q - flux composites Ahronov- Bohm effect : q

11 Physicl Anyons: Frctionl Quntum Hll DEG lrge B field (~ T) low temp (< K) gpped (incompressible) quntized filling frctions n h n R, R, xy xx m n e n n e n n e B / frctionlly chrged qusiprticles Abelin nyons t most p filling frctions non-abelin nyons in nd Lndu level, e.g. n 5/, /5, m

12 Xi, et l

13 non-abelin nyons Loclized topologicl chrge: b Non-locl collective topologicl chrge: (multiple vlues re possible) c b Fusion rules: ng. mom. nlog: c b c N c b c

14 Ising nyons -n -n 5 5 FQH - Kitev honeycomb, Topologicl Fusion rules : I (Moore- Red `9) nd other LL FQH?(PB nd Slingerln d `7) chrge I topologicl types: I,, insultors, I ruthentes? I I I I

15 Fiboncci -n 5 - string nets?(levin - Wen `4, Prticle FQH? types: Fusion rules : nyons (Red - Rezyi`98) I, Fendley et.l. `8) I I I I I

16 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) 3 4 c c Topologicl Protection! 3 4 Ising: Fib:, c I, c, c I, c

17 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) time Ising: not quite (must be supplemented) Fib: yes! (Bonesteel, et. l.) Is briding computtionlly universl?

18 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) time (Bonesteel, et. l.) Topologicl Chrge Mesurement

19 Topologicl Chrge Mesurement (mesures nyonic stte), ; c, ; c c c c c

20 Topologicl Chrge Mesurement e.g. FQH double point contct interferometer

21 FQH interferometer Willett, et. l. `8 for n5/ (lso progress by: Mrcus, Eisenstein, Kng, Heiblum, Goldmn, etc.)

22 Quntum Stte Teleporttion (for spin ½ systems) Entnglement Resource: mximlly entngled Bell sttes ( ( ( i ( 3,,,3

23 Quntum Stte Teleporttion Entnglement Resource: mximlly entngled Bell pir, ; (for spin ½ systems) Wnt to teleport: Form: 3 nd perform mesurement on spins

24 Mesurement Quntum Stte Teleporttion (for spin ½ systems) () 3

25 Mesurement Quntum Stte Teleporttion (for spin ½ systems) () : 3 3 Now send two bits of clssicl info (the mesurement result ) from Alice to Bob nd fix the stte by pplying the trnsformtion to spin 3

26 Mesurement Quntum Stte Teleporttion (for spin ½ systems) (3) () : 3 3 Now send two bits of clssicl info (the mesurement result ) from Alice to Bob nd fix the stte by pplying the trnsformtion to spin 3

27 Quntum Stte Teleporttion (for spin ½ systems) Alterntive fix : Recombine nd mesure the stte of spins 3 (3) n () : 3 3

28 Quntum Stte Teleporttion (for spin ½ systems) Alterntive fix : Recombine nd mesure the stte of spins 3 Then try gin: n (3) n () : n 3 n 3 n If mesurement outcome is If not REPEAT. n then STOP! ( success )

29 Forced Quntum Stte Teleporttion Mesurement (for spin ½ systems) ( n ( ( 3 n 3

30 Forced Quntum Stte Teleporttion Mesurement (for spin ½ systems) ( ( ( 3 ( 3 ( n n n ( : 3 3 Success occurs with probbility 4 for ech repet try.

31 Anyonic Stte Teleporttion Entnglement Resource: mximlly entngled nyon pir,; Wnt to teleport: Form:,; 3 nd perform Forced Mesurement on nyons

32 Anyonic Stte Teleporttion Forced Mesurement ( n ( ( 3 n,; 3

33 Forced Mesurement Anyonic Stte Teleporttion ( ( ( 3 ( 3 ( n n n ( :,; 3, ; 3 Success occurs with probbility d for ech repet try.

34 Wht good is this if we wnt to brid computtionl nyons? R

35 Use mximlly entngled pir nd forced mesurements for series of teleporttions

36 Use mximlly entngled pir nd forced mesurements for series of teleporttions

37 Use mximlly entngled pir nd forced mesurements for series of teleporttions

38 Use mximlly entngled pir nd forced mesurements for series of teleporttions

39 Mesurement Simulted Briding! R (4) (3) (34) (3)

40 in FQH, for exmple

41 in FQH, for exmple

42 in FQH, for exmple

43 in FQH, for exmple

44 Topologicl Quntum Computtion time mesurement simulted briding Topologicl Chrge Mesurement

45 Mesurement-Only Topologicl Quntum Computtion Topologicl Chrge Mesurement Topologicl Chrge Mesurement

46 Conclusion Anyons could provide quntum computer. Teleporttion hs nyonic counterprt. Bounded, dptive, non-demolitionl mesurements cn generte the briding trnsformtions used in TQC. Sttionry nyons hopefully mkes life esier for experimentl reliztion. FQH interferometer technology is rpidly progressing.