Measurement-Only Topological Quantum Computation
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1 Mesurement-Only Topologicl Quntum Computtion Prs Bonderson Microsoft Sttion Q DAS Theoreticl Physics Seminr August 21, 2008 work done in collbortion with: Mike Freedmn nd Chetn Nyk rxiv: (PRL 08) nd rxiv:
2 ntroduction Non-Abelin nyons probbly 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? They could hve ppliction in quntum computtion, providing nturlly ( topologiclly protected ) fulttolernt hrdwre. Assuming we hve them t our disposl, wht opertions re necessry to implement topologicl quntum computtion?
3 Anyon Models (unitry brided tensor ctegories) Describe qusiprticle briding sttistics in gpped two dimensionl systems. Finite set C of nyonic chrges:, b, c Unique vcuum chrge, denoted hs trivil fusion nd briding with ll prticles. Fusion rules: b = c N c b c c N b Fusion multiplicities re integers specifying the c dimension of the fusion nd splitting spces V, V b b c
4 Hilbert spce construct from stte vectors ssocited with fusion/splitting chnnels of nyons. Expressed digrmmticlly: nner 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 sing nyonsor SU -n = -n = FQH Prticle types: Fusion rules :, ( 2 nd other 2LL FQH?(PB nd Slingerln d `07) - Kitev honeycomb, (Moore- Red `91), 2 topologicl (.k.. 0, insultors, 1 2, 1) ruthentes?
8 Fiboncci -n = 12 5 Prticle types: Fusion rules :, ( nyonsor SO 3 FQH? (Red - Rezyi`98) - string nets?(levin - Wen `04, 3 (.k.. 0, Fendley et.l. 1) `08)
9 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) 0 c 1 0 c1 Topologicl Protection! sing: = 0 1, c =, c = Fib: = 0 1, c =, c =
10 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) c c1 time (Bonesteel, et. l.) s briding computtionlly universl? sing: not quite (must be supplemented) Fib: yes!
11 Topologicl Quntum Computtion (Kitev, Preskill, Freedmn, Lrsen, Wng) c c1 time (Bonesteel, et. l.) Topologicl Chrge Mesurement
12 Topologicl Chrge Mesurement Projective (von Neumnn) e.g. loop opertor mesurements in lttice models, energy splitting mesurement =, b; c, b; c c = c b c c
13 Topologicl Chrge Mesurement nterferometric (PB, Shtengel, Slingerlnd `07) e.g. 2PC FQH, nd Anyonic Mch-Zehnder (idelized, not FQH) Asymptoticlly chrcterized s projection of the trget s nyonic chrge AND decoherence of nyonic chrge entnglement between the interior nd exterior of the trget region. (more lter; ignore for now)
14 Anyonic Stte Teleporttion (for projective mesurement) Entnglement Resource: mximlly entngled nyon pir, ; = Wnt to teleport: = Form:, = 1 ; 23 nd perform Forced Mesurement on nyons 12
15 Forced Mesurement (projective) ( 12 : Anyonic Stte Teleporttion ( 12 ( 23 f2 f 2 f 2 ( 12 e1 e 1 e 1 ; 1, 23 =
16 Anyonic Stte Teleporttion Forced Mesurement (projective) ( 12 : ( 12 ( 23 f2 f 2 ( 12 e1 e 1
17 Forced Mesurement (projective) Anyonic Stte Teleporttion ( 12 ( : 12 : ; 1, 23, ; 12 3 = Success occurs with probbility 1 2 d for ech repet try.
18 Wht good is this if we wnt to brid computtionl nyons? R =
19 Use mximlly entngled pir nd forced mesurements for series of teleporttions (23) (34) (13) (23)
20 Use mximlly entngled pir nd forced mesurements for series of teleporttions (23) (34) (13) (23)
21 Use mximlly entngled pir nd forced mesurements for series of teleporttions (23) (34) (13) (23)
22 Use mximlly entngled pir nd forced mesurements for series of teleporttions (23) (34) (13) (23)
23 Mesurement Simulted Briding! R (14) (23) (34) (13) (23) =
24 in FQH, for exmple
25 in FQH, for exmple
26 in FQH, for exmple
27 Topologicl Quntum Computtion c c1 time mesurement simulted briding Topologicl Chrge Mesurement
28 Mesurement-Only Topologicl Quntum Computtion c c1 Topologicl Chrge Mesurement Topologicl Chrge Mesurement
29 Topologicl Chrge Mesurement nterferometric (PB, Shtengel, Slingerlnd `07) e.g. 2PC FQH, nd Anyonic Mch-Zehnder (idelized, not FQH) Asymptoticlly chrcterized s projection of the trget s nyonic chrge AND decoherence of nyonic chrge entnglement between the interior nd exterior of the trget region.
30 nterferometricl Decoherence of Anyonic Chrge Entnglement =, b; c, b; c = For inside the interferometer nd b outside: D int : = c
31 nterferometricl Decoherence sing: D : = int D : = int
32 nterferometricl Decoherence Fiboncci: D : = int
33 Mesurement Generted Briding! Using nterferometric 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 sing model TQC qubits, interferometric mesurements re projective.
34 sing vs Fiboncci (in FQH) Briding not universl (needs one gte supplement) 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 nd (precise phse clibrtion) Briding is universl (needs one gte supplement) D n=12/5 ~ 70 mk Brids = Unnturl gtes (see Bonesteel, et. l.) nherent lekge errors (from entngling gtes) nterferometricl MOTQC (2,4,8 nyon mesurements) Robust mesurement distinguishing nd (mplitude of interference)
35 Conclusion Quntum stte teleporttion nd entnglement resources hve nyonic counterprts. Bounded, dptive, non-demolitionl mesurements cn generte the briding trnsformtions used in TQC. Sttionry computtionl nyons hopefully mkes life esier for experimentl reliztion. Experimentl reliztion of FQH double pointcontct interferometers is t hnd.
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