Measurement-Only Topological Quantum Computation
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1 Mesurement-Only Topologicl Quntum Computtion Prs Bonderson Microsoft Sttion Q UIUC Workshop on Topologicl Phses of Mtter October 24, 2008 work done in collbortion with: Mike Freedmn nd Chetn Nyk rxiv: (PRL 08) nd rxiv:
2 Introduction Non-Abelin nyons re believed to exist in certin gpped two dimensionl systems: - Frctionl Quntum Hll Effect (n=5/2, 12/5,?) - 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 non-abelin nyons Loclized topologicl chrge: b Non-locl collective topologicl chrge: (multiple vlues re possible) c b Fusion rules: ng. mom. nlogy: c b = c 1 1 = N c b c
4 Hilbert spce construct from stte vectors ssocited with fusion/splitting chnnels of nyons. Expressed digrmmticlly: Inner product: = cc'
5 Associtivity of fusing/splitting more thn two nyons is specified by the unitry F-moves:
6 Briding b R = = c b R c Cn be non-abelin if there re multiple fusion chnnels c U R
7 Physicl Anyons: Frctionl Quntum Hll 2DEG lrge B field (~ 10T) low temp (< 1K) gpped (incompressible) quntized filling frctions n h n R =, R =, xy 2 xx m 1 = n e 0 n = n e n = n e B / 0 frctionlly chrged qusiprticles Abelin nyons t most p filling frctions = non-abelin nyons in 2 nd Lndu level, e.g. n= 5/2, 12/5, m
8 Xi, et l
9 Ising nyons -n = -n = FQH - Kitev honeycomb, Topologicl Fusion rules : I (Moore- Red `91) nd other 2LL FQH?(PB nd Slingerln d `07) chrge I topologicl types: I,, insultors, I ruthentes? I I = I = I
10 Fiboncci -n = string nets?(levin - Wen `04, Prticle FQH? types: Fusion rules : nyons (Red - Rezyi`98) I, Fendley et.l. `08) I I I I = I
11 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) c0 c1 Topologicl Protection! Ising: Fib: = 0 1, c = I, c = = 0 1, c = I, c =
12 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) time Ising: not quite (must be supplemented) Fib: yes! (Bonesteel, et. l.) Is briding computtionlly universl?
13 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) time (Bonesteel, et. l.) Topologicl Chrge Mesurement
14 Topologicl Chrge Mesurement (mesures nyonic stte) 1 2 =, ; c, ; c c = c 1 2 c c 1 2 c
15 Topologicl Chrge Mesurement e.g. FQH double point contct interferometer
16 FQH interferometer Willett, et. l. for n=5/2 (lso progress by: Mrcus, Eisenstein, Kng, Heiblum, Goldmn, etc.)
17 Anyonic Stte Teleporttion Entnglement Resource: mximlly entngled nyon pir, ; I = Wnt to teleport: = Form:, I = 1 ; 23 nd perform Forced Mesurement on nyons 12
18 Anyonic Stte Teleporttion Forced Mesurement (projective) ( 12 : I ( 12 I ( 23 f2 f 2 I f 2 1 e1 ( 12 e e 1 ; 1, I 23 =
19 Forced Mesurement (projective) ( 12 : I Anyonic Stte Teleporttion ( 12 I ( 23 f2 f 2 ( 12 e1 e 1
20 Forced Mesurement (projective) Anyonic Stte Teleporttion ( 12 ( : 12 : I I ; 1, I 23, ; I 12 3 = Success occurs with probbility 1 2 d for ech repet try.
21 Wht good is this if we wnt to brid computtionl nyons? R =
22 Use mximlly entngled pir nd forced mesurements for series of teleporttions 0
23 Use mximlly entngled pir nd forced mesurements for series of teleporttions 0
24 Use mximlly entngled pir nd forced mesurements for series of teleporttions 0
25 Use mximlly entngled pir nd forced mesurements for series of teleporttions 0
26 Use mximlly entngled pir nd forced mesurements for series of teleporttions 0
27 Use mximlly entngled pir nd forced mesurements for series of teleporttions 0
28 Mesurement Simulted Briding! R (14) (23) 0 (34) 0 (13) 0
29 in FQH, for exmple
30 in FQH, for exmple
31 in FQH, for exmple
32 in FQH, for exmple
33 Mesurement-Only Topologicl Quntum Computtion Topologicl Chrge Mesurement mesurement simulted briding Topologicl Chrge Mesurement
34 Mesurement Generted Briding! Using Interferometric Mesurements is similr but more complicted, requiring the density mtrix description. The resulting forced mesurement procedure must include n dditionl mesurement (of 8 or fewer nyons, i.e. still bounded size) in ech teleporttion ttempt to ensure the overll chrge of the topologicl qubits being cted upon remins trivil. Note: For the Ising model TQC qubits, interferometric mesurements re projective.
35 Ising vs Fiboncci (in FQH) Briding not universl (needs one gte supplement) Almost certinly in FQH D n=5/2 ~ 600 mk Brids = Nturl gtes (briding = Clifford group) No lekge from briding (from ny gtes) Projective MOTQC (2 nyon mesurements) Mesurement difficulty distinguishing I nd (precise phse clibrtion) Briding is universl (needs one gte supplement) Mybe not in FQH D n=12/5 ~ 70 mk Brids = Unnturl gtes (see Bonesteel, et. l.) Inherent lekge errors (from entngling gtes) Interferometricl MOTQC (2,4,8 nyon mesurements) Robust mesurement distinguishing I nd (mplitude of interference)
36 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.
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