Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Twists in Topological Codes Bombin, Hector Jul 25, 2010

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Twists in  Topological Codes Héctor Bombín Perimeter Institute  Outline Motivation Anyons and twists Twists in toric codes Twists in topological subsystem codes  Motivation Quantum error correction attempts to create faithful quantum channels from noisy ones  Encoding  Decoding  Noisy channel  Noiseless channel  Typically this involves encoding in a subspace  PACS numbers: 05.30.Pr, 03.67.Lx, 73  Motivation The code subspace can be defined in terms of commuting observables: check operators (CO)  Ci |ψ = ci |ψ Errors typically change CO values: CO measurement → error syndrome → → compute mostzprobablezerror →correct  σ1 ⊗ σ2  (a) Motivation  (b)  (d)  (c)  In some settings locality is crucial  Ci = ci Topological codes have geometrically Ci local =c cCOs i Topological subsystem codes (TSC): a Just 2-local measurements!  2  Z1 Z2 Z1IZ2  3  Y2 X3 Y2 X3  1 (a) r  X  (b)  Motivation Fault-tolerance in topological memories: repeatedly measure error syndrome → → keep track of errors  How to compute? Transversal gates (color codes) Boundaries & code deformation NOT possible for TSCs!!!  Motivation Topological codes  VS  Topological order  Code subspace  Ground subspace  Error correction  Energy gap  Error syndrome  Excitation configuration  In 2D excitations are very special: ANYONS. Using anyon symmetries → All CLIFFORD gates by code deformation on TSCs!!!  quantum computation [? ? ]. PACS numbers: 05.30.Pr,In03.67.Lx, condensed73.43.Cd matter, anyons emerge as excitations in Ci topological = ci systems that exhibit order [? ]. A possible y suitx way to obtain these exotic is by engineering y phases x σ2 ⊗ σ3 able on2lattice 3 zsystems [? 2? ? ? ?3 z z Hamiltonians z spin ]. Indeed, implementations on optical lattices have been 2 2 1 1z z σ1 ⊗ σ2 the anyon models that approposed [? ]. Unfortunately, Ci = cipear in simple models are not computationally 1powerful. =  rules unchang symmetry s, 1 a open curve, (a to s”. Ideally only its endp I ular, transpo r r g r r the gc changes of s. Our aim 2D: statistics beyond bosons and fermions b b g defects, ical t (a) b g y x x as anyons” an σ2y ⊗ σy3x r dently studie 3 3 2 2 Q ∈ {a, b, . . . }precedent are (c t z z σ1 ⊗ σ2 r r g g b models [? r], = monodromy, elec g g g b b Q r b exchange y x perturbatio Ingredients of an anyon model: σ2 ⊗ σ3 Rather tha m Q ∈ {a, b, . . . } quantum c will focus on c In conde model, and a a b a b a b (c) e + q2 + Q × Q = q . . . haveth anyons 1 systems = j jp Qj jp that twists b way to obt (a) (b) (c) (d) putationally  σ ⊗σ Anyons σ ⊗ σ Y2 X σ 3⊗ σ  σ ⊗σ =  = σ ⊗σ  =  Q ∈ {a,= b, . . . }  le statistics are particularly rich in two s Ind s, where Qbeyond usual ∈ {a, b, . . .the Q } ∈Q {a, b, . . . } fermions]. an propo able Hamil st more generally anyons (see [? ] fo ]. Indeed, implementations Q ∈ {a, b, . . . } |ψ −→ ? Q × Q = q + q + . . . ]. spatia Indeed, e statistics Particle arestatistics particularly statisticsare areparticularly rich particularly in two spatial richinin ditwospatial pear dii Particle rich two proposed [? ]. Unfortunate proposed [? , where mensions, beyond where the usual beyond fermions the usual and fermions bosons and bos Top. charges Fusion rules Braiding rules Particle statistics particularly rich in two spatial diIn th mensions, where beyond the usual fermions and boso of the basic references). Anyonic stat statistics are particularly rich inare two spatial dipear in sim pear in simple models n Particle statistics are particularly rich are in tw mensions, where beyond the bosons usual fermions and bosons In this pap where beyond the usual fermions and st more there generally exist more anyons generally (see [? anyons ] for (see a com[?usual ] for for acom co In(see this the paper we address a tation there exist more generally anyons [? ] a mensions, where beyond fermions there exist more generally anyons (see [? ] for a comtationally enough to give rise to the notion of to more generally anyons (see [? ] for a comtationally interesting anyo there exist more generally anyons (see [? as of the pilation basic ofthe the basic Anyonic references). statistics Anyonic are statistics pilation of the basic references). Anyonic statistics are with pilation ofreferences). basic references). Anyonic statistics with very 1  2  q q q q q q 1 2 Q q× Q = q + q + . . . 1 2 3 Q × Q = q + Q×Q = + q + . . . 1 2 1 3q2 + . . . 1 2  form Particle statistics are particularly rich in two spat Particle statistics are particularly rich in two spatial diq q q of b 1 2 3 Topological QC wherethe beyond usual and fermions and b mensions,mensions, where beyond usual the fermions bosons per |ψ −→ ?|ψ −→ ?|ψ −→ ? Qubits encoded on fusion channels qua I .q1.q.2 q3 q1 q2 q3 q1 q2 q3 syst way Compute = Braid Measure = Fuse icle statistics are particularly rich in two spatial di Particle statistics are particularly rich in two stics are particularly rich in two spatial diare particularly rich in two spatialable d ns, where the beyond the usualthe fermions andm boson where beyond usual fermions an e mensions, beyond usual fermions and bosons ]. I eyond the usual fermions and boson t pro e enerally anyons (see [? ] for a com pea (a) (b) (c) a x b = c (d) (a) (b) (c) (d) BUT: In abelian models → trivial code c references). Anyonic statistics Inart tati give rise to the notion of topologica  ⊗ of their topological ⊗ ome permutations charges. Symmetries. . . under monodromy, transform anyons according Anyon symmetry: charge permutation producing = = (a) defects in the toric code model, showing that a pro an equivalent anyon model ame outcome as if they were Ising anyons. These id Q ∈ {a, b, . . . } Q ∈ {a, b, . . . } ical codes. q −→ π(q) y σ2  y σ2  x σ3  x σ3  Q Q Imagine ‘cutting’ the anyons’ 2D world and gluing  statistics are particularly rich in two it again up to a symmetry Q×Q =q +q + Q .×. .Q = q + q + . . . 1 where beyond the usual fermions(a)a |ψ −→ ? |ψ −→ ? t more generally anyons (see [? ] g r q q q q q Anyonic sta f the basic qreferences). 1  2  1  2  2  g  r  1 2 3  1 2 3  g  b  b  Q × Q = q1 + q2 + . . .  |ψ −→ ? harges.Q One × Q = q + q + . . . 1 2 Twists . . . . . . cording to a |ψ −→ ? Across the cut, charges change: hat a process q1 q2 q3 . These ideas π(q) q q−→ π(q) |ψq −→ −→ ? 1 q2 q3  Topologically, the cut location isrich unphysical. icle statistics are particularly in two Particle statistics are particularly in spatial two sp . . . rich Endpoints are meaningful: under monodromy ns, where beyond the usual fermions and and bos nsions, where beyond the usual fermions ... q q q they permute charges → TWISTS 1 2 3 xistexist moremore generally anyons (see (see [? ] [?for] aforco ere generally anyons q −→ π(q) nation of the basicbasic references). Anyonic statistics of the references). Anyonic statis q −→ π(q) x enough to give rise rise to the notion of topolog mplex enough to give to the notion of top Particle statistics are particularly rich in tw . . . m computation (TQC) [? ?[? ? ? ], ?where com antum computation (TQC) ], where mensions, where the usual Particle statistics arebeyond particularly rich infermions two spa are carried out by braiding and fusing anyo  emR(b) me = R e R(b)e= R mR m= −1 (a) H. Bombin R em mer e e m m Bombin r r g r b 31 St. N.,b Waterloo, Ontario N2L Canada g2Y5, r gg b r Waterloo, g Canada R = R = 1 R = −1 b g 1aroline r g r Caroline g g r g bSt. bN., r b ee mm Ontario N2L 2Y5, b Rree r= gRmm = 1 R = −1 g g r g b g r b g r g b g b r r b se b g(b) rb g g g RembrRme rb r br(a)g g = R R = R R = −1 (1) b e e m m g g {1, e, m, } rR gRemr R r bR b me g= Rof b b = R =}charges. −1rOneg One gm, b (1) r some permutations their topological r {1, e, e e m m b me permutations of their topological charges. fo g b r g r b b g r b b g g gr to ba b r g r b b b at,bb under monodromy, transform anyons according g g g r g g g b r r b b b b gmonodromy, nder transform anyons according to a g of (d) g Example: g double b r Z2r (toric b code) b g quantum of (e) (c) r b g g g g b gb r g r g bb b r g b g g r b m, } that re,that br a g defects ch thecode toric code showing a process g model, r model, g in the b {1, b in g g ects toric showing process g g r b g b r b pe Charges: {1,= e,were m, anyons. }ee××anyons. × m = =mm ideas m×ideas × ==e e he same outcome as ifwere they Ising These ee × m = m me outcome as if they Ising These g (d) r b b (c) (e) qu g g g r r (c) b b blogical codes. (d) (e) l codes. (d) (e) Fusion:(d)e × m = em×× = m m × =1 e e × e = × m = × g = g (e) (c) (d) e × e = m m = × = 1 r b (e) Particle statistics are particularly = rich − in two spatial die×e=m×m= × =1 (2) sy ensions, (1) where beyond the usual fermions and bosons = − (e) (c) (d) w ere exist more generally anyons (see [? ] for a comParticle statistics are particularly rich in two spatial di= − Braiding: = − e, m → bosons → bosons e, m → bosons → fermion = − = − ab lation of the basic references). arebosons mensions, where 1 beyond theq usual fermions and = q Anyonic = = 1statistics theretoexist more generally anyonsof(see [? ] for a com]. mplex enough give rise to the notion topological 1=q=q =1 =− pilation of the basic references). Anyonic statistics are pr uantum 1 computation (TQC) [? ? ? ], where compu1 = q = q = 1 = q = q = 1 1 = q = q = 1 1=q=q =1 pe tions are carried out statistics by braiding and fusing anyons, icle statistics are particularly rich in two spatial diParticle are particularly rich in two spa Nontrivial symmetry: e ↔ m realization? Particle statistics are particularly rich in two spatial dins, where beyond the usual fermions and bosons Particle statistics particularly rich in two spatial dimensions, where beyond the usual fermions and b tistics are particularly rich in two spatial ditistics particularly rich in two spatial di(2) xist more generally anyons (see [? ]bosons for a and comnsions, where beyond the usual usual fermions and bosons there exist more generally anyons (see [? ] for a ensions, where fermions bosons ere the usual fermions and bosons ere beyond the usual fermions and Particle statistics are particularly rich in two spati n of the basic references). Anyonic statistics are  Twists  r  Z gg  −1  Z  =(1) −1  1  bb  (c)  bb  rr  bb  gg  gg  (c)  (d)  bb  rr  gg  (d)  Toric code 1=q=q  (1)  (1) (1)X  X  Z  (c) (c)  Z  =−  rb b b r r bb b b gg gg (e) r r b b g g(e)  (d) (d)  =−  X  Z  ==−− lattice Qubits form a square 1 = q = q =1 1= q = q = 1  b  =1  (e) (e)  X  Z X y Z e ↔ m 4-local check operators at plaquettes 11==qq==qq ==11  × =e  e↔m  e↔m  Z  X  X  Z  prec ee↔ ↔mm mod X Z XAk :=ZXk Zk+i AkX:= Xk Zk+i . k+i+j Zk+j . k+i+j k+jX (2) (2) mon ion X(a) Z Ck := Xk Zk+i Xk+i+j Z . k+j (b) XX ZZ AAkk:= :=XXkkZZk+i XXk+i+j ZZk+j .. k+i k+i+j k+j exch R will Hamiltonian: mod H := − Ck , anyo k Z that Excitations live in plaquettes puta Z X Z allow ParticleXstatistics are particularly rich in two spatial d X X Z X Z Z to o  1 (2) ee  (2)  g b = r gg bgbg g b b bg r R br r −1 g r r 1−1 R = −1 = b= g g b= b gg b b−1 =R −1 Rmm R Rm (1) m(c) −1r r b(c) −1 −1  =(1) =−1−1  m, m, }}  (1) (1)  = m × == m e= e (2) m 1 = × × e= × = (2) (2)e  g gr b bbb r rg g g r b bb g r r r g g g gb b b r r bg r rg gg g rb b b g g r r bb b b gg gg gg rr bb gg (e) r r b b g g(e) (d) (d) b  g  (c) (1) (1)  Toric code  (1)  (c) (c)  =−  (d)  ==−−  (e) (e)  (d) (d)  (1) =− =− =− String operators create/destroy excitations at 11==qq==qq ==11 1 = q = q = 1 ==−− their endpoints 1 = q = q =1 1= q = q = 1  m m× × = = ee (2) = = 11 (2)  1=q=q =1 ↔mm e↔m ee↔  11==qq==qq ==11 e↔m e↔m  prece (2) ee↔ ↔mm mode X Z XAk :=ZXk Zk+i AkX:= Xk Zk+i . k+i+j Zk+j . k+i+j k+jX e ↔ m mono on XAk :=ZZXk Zk+i := X X Z X AAkX := X Z . kk Z k+i k+j k k+i+j k+X 1 (2) XX ZZ AAkk:= :=XXkkZZk+i XXk+i+j ZZk+j .. k+i k+i+j k+j exch mion → →fermion fermion Ra X Z Ak := Xk Zk+i Xk+i+jwill Zk+ ion mode anyo Two types of excitations: e (light) and m (dark) Z that puta Z X Z allow X  Z  X  Z  X  Z  X  g  bg  bb  b r  r b b b g g g g r g g g g b b r r g r br b g g b bg b b rb b bgr g gg b b r g g g(e) g r rb(d) b(e) elec (c) (c) (d) (d) (e) exchange (d) (c)  (c) q−→ −→π(q) π(q) = ==1−1 R = −1 (c) Rather tha R m −1 (d) (e) (e)(d) (c) (c) (d) (d) (e) (c)(c) (d) will focus on R = −1 = R R (1) = −1 (1) (1) −1 emR m e m m (1) = −= − =− =− −1 −1 ==1 1 RR ==−1 m (1) model, and a =− −1 Re −1 (1) −1 (1) = ==RR RRm (1) ==−1 (1) e m meR mm anyons have n = − = − = − = − {1, e, m, } To get twists, we simply add dislocations 1= = q twists =1 1=q=q = 11q==qq==q 1=1 1= qthat be  Toric code twists  putationally {1, , e,e, m,m,} } 1 = q1==1qq== 1 =q q== q= q 1= 11 1 == q= q q= = 11 m e ××m = =m e= e m × = e × e ↔e m e allow ↔ m univers e↔m ↔m to overcome t m =× 1=m 1 (2) (2) × (2) 1= × =(2) e×× = × e===m em mm ×× ==e e e↔m e ↔e m e et ↔ m also try ↔m e↔ mal. (2) Z (2) A :=Z X Zk+i Zk+i .Ak+i+j . k+i+ XAX Z XZ AkXX :=kk+i+j X X := X Zk+j . X k := k Zk+i k k+j k+i+j k+j kZ kX k Zk+i m× × =(2) =1 1 X (2) 1×mm== ×(2) abelian system →rmion fermion → fermion ons philosophy. := Xk+i Z ZZk+i+j X AZ := Z .Zk+i Xk+i+j Z := XZkk+i Z Xk+i+j Zk+j := .XZkk+i Xk+i+ X ZX XY Z ZCkA:= X .X k k kX k+i k+i+j kAk+j k+j k kA k We remark mion →→fermion nns ons fermion  be in terms of (b) (a) Twists can be locally created in PAIRS only in the closely ? ]. Anyon mod characterized plex enough to give rise to the notion of topological  X  Z  kY  k k k+i k+ C := Xk+i+j k Zk+i X k e ↔em↔ m  ocess e ↔ m X Z Y C := X Z X Z . k k k+i k+i+j k+j e ↔ m Generalized charges H H := := − − C C , , H := − C , H := − C , H := − C , k k k k k ideas − C , ) k H := − (1) X X Z Z Y kH Yk k:= C := X Xk , Z Z. C := X Z ZX C k  (1)  kk  k  k k+i k+i+j k+i k+i+j k+j EVEN charges k X number Z of twists Y → 4Cpossible := X k k Zk+ikXk+i+j Zk+ X Z H := Y − CkC:= k , Xk Zk+i Xk+i+ k  Sm Sm − 1 1 1 1 1 e 1 m e m S− − 1 1 1 e m 1 1 e m SeSeSeSeSSm 1− 1 1 e m k eSm m− H −:= − Ck , Ck , H := Se SS − 1 1 1 e m m H := − C , +1 S − 1 1 1 e m +1 -1 -1 k e m k k (1) )rticle H := −in Cspatial ,di-di-di-diParticle statistics are particularly rich in two dikspatial Particle Particle statistics statistics are are particularly particularly rich in two two spatial Particle statistics are particularly rich in two spatial statistics are particularly rich inrich two spatial k (1) S S − 1 1 1 e m +1 -1 +1 -1 e m kand mensions, where beyond the usual fermions and bosons mensions, mensions, where where beyond beyond the the usual usual fermions fermions and bosons bosons nsions, where beyond the usual fermions and bosons ions, where beyond the usual fermions and bosons (1) Particle statistics are particularly rich in spa Particle statistics are particularlytwo rich in S S − 1 1 1 e m there exist more generally anyons (see [? ] for a comS S − 1 1 1 e m here there exist exist more more generally generally anyons anyons (see (see [? [? ] ] for for a a comcome m re exist more generally anyons (see [? ] for a come m exist more generally anyons (see [? ] for a commensions, where beyond the usual fermions and Particle statistics are particularly rich in two spati mensions, where beyond the usual fermi )ation S S − 1 1 1 e m e m (2) pilation of the basic references). Anyonic statistics are ilation pilation ofthe of the the basic basic references). references). Anyonic Anyonic statistics statistics are are ODD number of twists → 2 possible charges ofthe basic references). Anyonic statistics are ion ofthere basic references). Anyonic statistics are S S − 1 1 1 e m e m exist more generally anyons (see [? ] for mensions, where beyond the usual fermions and bo there exist more generally anyons (see [ (2) complex enough to give rise tothe the notion oftopological topological omplex complex enough enough to to give give rise rise to to the notion notion of of topological mplex enough to give rise to the notion of topological plex enough to give rise to the notion of topological there pilation exist more (see for a pilation of the basic references). statist Sbasic σreferences). σ− [? ]Anyon Sthe σ+anyons ofgenerally + σ−Anyonic S σ σ + −notion statistic pilation of the basic references). Anyonic complexcomplex enough to give rise to the of topo enough to rise to the notio S give σ +i σ-i + − complex enough to give rise to the notion of topolo Particle statistics are particularly rich in two s Particle statistics are particularly rich in two spati ) (2) Particle statistics are particularly rich in two spa (2) mensions, mensions, where beyond the usual fermions an where beyond the usual fermions and b Particle statistics are particularly rich in tw (2)mensions, where beyond the usual fermions and  k − H := H := − Ck , Ck ,  (1)  1)  H := − C , Fusion rules k H := − C , Se Sm −1 k11em k  k  k  k  SeSm Sm− 1 − 11 1 e m1 1 e m Twists are sinks for fermions: S σ+ σ− Se fusion 1rich e min two spati Non-abelian rules! Particle are−particularly Se statistics Sm Sm 1 − 1 1 1 e1m S σ σ + − mensions, where beyond the usual fermions and bo (2) Particle are particularly rich in two spatial σ±statistics × σstatistics = 1 + σ × σ = e + m Particle are particularly rich in two spatial there exist more generally anyons (see [? ] fordia ± ± ∓ mensions, where beyond the usual fermions and bos mensions, where beyond the usual fermions and bosons pilation of the basic references). Anyonic statistic 2) σ±±= 1 +σ± × eσ±=×σ± σ∓×=me = +m σ±σ× σ∓ (3) ± ×=σ there more generally anyons (see ] offor a co there existexist more generally anyons (see [?notion ][? for a topolo comcomplex enough to give rise to the σ × =σ σ × e = σ × m = σ (3) ± ± ± ± ∓ pilation of the basic references). Anyonic statistics pilation ofstatistics the basic references). Anyonic statistics are Particle are particularly rich in two spatial diWe recover Ising rules: complex enough to give rise to the notion of topolog complex enough to give rise to the notion of topological mensions, where beyond the usual fermions and bosons σ ×more σ = 1generally +ψ σ anyons × ψ = σ(see ψ = 1a com(4) there exist [?×]ψ for pilation of the basic references). Anyonic statistics are Particle statistics are particularly rich in two spatial dicomplex enough to give rise to the notion of topological mensions, where beyond the usual fermions and bosons Se  S σ σ + − cj ckk k+ ck cj = 2δjk c1 c2 c3 em SeSeSe SmSmSm − − 1 − 1 1 1 11e11m e1m c c + c c = 2δ c c c j k k j jk 1 2 3 S − 1 1 1 e m m Majorana operators σ × σ = 1 + σ × σ = e + S S − 1 1 1 e m S S e m ± ± ± ∓ e m } Q σ × =σ σ × e = σ × m = ± ± ± ± ×σ σcan + σ∓ = e be expressed inσ terms SσS Sstring σσ±σops σ= σ 1σ S σAll closed ± ± ×of S  (b)  + σ+  − σ−  + + + − − −  a set of open σ string × ops =σ→ Majorana σ operators ×e=σ  S c × m 2δ ± ± ± ± 1 c2cc 3 (b) =jk2δcjk c c 1 2 3 2δ +k c+jk c= 2δ 2δjk c1jk c21cc321c32 c3 j = jk c= jk σ × σ = 1 + ψ σ × ψ = σ ψ × σ× = 1 + σ × σ = e + m σ∓ e +. m ± ∓q ±× ×σ±Q= 1 += q±1σ+ . . cj ck + ck cj = 2δjk c1 c2 c3 2 =+ =σ σ × e = σ × m = σ (3) × =σ σ × e = σ × m = σ (3) ± ± ± ∓ ± ± ± ∓ σ × σ = 1 + σ × σ = e + m σ σ × × σ σ = = 1 + 1 + σ σ × × σ σ = = e + e + m m ± ± ± ∓ ± ± ± ± ± ± ∓ ∓ σ×σ =1+ψ σ×ψ =σ ψ + ... Braiding is also Ising-like → 1-qubit Clifford gates σ × σ = 1 + ± ± σ× ×=σ =σ σ× × e±= σc× × m = σ (3 σ σ × e e = σ σ × m m = = σ σ (3) (3) ±=σ ± σ±σ× ±= ± ∓ ± ± ± ± ± ± ∓ ∓ → c j+1 σ1=+1ψ+ ψ σ ×σψ×=ψ σ= σ ψ × ψψ ×= ψ= 1 1 (4)(4) j σ± × =σ± |ψ −→ ? m cj → cj+1 cj c→ cj+1 c → j j+1 × σ+ = σ× ψ ψ× ψ = 1 (4 × σ× σσ = σ= 1 1+ ψ1 + ψ ψσ × σ× ψ = ψ= σ= σ σcj+1 ψ× ψ→ × = ψ = 1 1 (4) (4) −cj (e) σ×σ =1+ψ cj+1 → j−cj c → −c j+1 article statistics are particularly rich in two spatial ticle le statistics statistics particularly rich rich in in two two spatial spatial di-di-d c → −c q1 qare j+1 j 2 qare 3particularly (e) Particle statistics are p nsions, where beyond the usual fermions and boson ons, s, where where beyond beyond thethe usual usual fermions fermions and and bosons bosons m  e  j  j+1j  j+1j  j+1j  j+1 j (a)  j+1  (b)  c→ −c c→ →j −cj −c c→ −c cj+1 c→ −c j+1 j+1 j+1 j j+1 j j c → c k kccj+1 ccjj → c → c c → ccjj → ccjj → ccj+1 →ccj+1 c → c c → →ccj+1 → j j+1 j j+1j j+1j j+1 j+1 j+1 TSC twists cj+1 → −c cj+1 cj+1 cj+1 j → −c j → −c j → c−c j → −cj j+1 c → c c → c c → c c → c c → c k k k k k k k k k k Charges: {1, r, g, b} ccj+1 c−c c−c c−c →−c cj+1 →−c cj+1 →−c cj+1 →c−c −c cj+1 →−c −cjj jj → jj → jj → jj → j+1 → j+1 j+1 j+1 j+1 ck → ckinctoric ck ckbut →allcare → ckck → ck Like k →code, k ckfermions {1, r, g, r, b} {1, g,fermions b} {1, r, g, b} {1, r, g, b} {1, r, g, b} Symmetries: any permutation of Γ = {e, ζ , ζ , σ , σ , σ } + − r g b ) c(1) ckk → →cckkcckk → →cckkcckk → →cckkcckk → →cckkcckk → →cckk 2 {1, r, g, b} {1, r, g, b} {1, r, g, b} {1, r, g, {1, b} r, g, b} statistics are particularly rich in two spatial di))icle (1) (1) ζ{e, , ,ζrζσ− Γ =ζ+{e, ζ+{e, σ ,− σ Γ, ζ=− ,= ,+bσ,,}gσζ,r−σ, ,σ }gr,,σσbg}, σb } Γ = {e, ,Γ,σζ= ,ζΓ σ,{e, ,ζrσ + gσ + bσ r− g b, } ns, {1, where fermions and bosons b} {1, b} {1, r,r,g, {1, r,r,g, b} {1,r,r,g, g,beyond b} {1,r,r,g, g,the b} {1,usual g,b} b} {1, g,{1, b} r, {1, r,g, g,b} b} =exist {e, ζ Γ = , ζ {e, , σ ζ Γ , σ = , ζ , {e, σ , σ } ζ Γ , σ = , ζ , {e, σ , σ ζ } , σ , ζ , σ , σ } , σ , σ } Γ = {e, ζ , ζ , σ , σ , σ more generally anyons (see [? ] for a com+ − r + g − b r + g − b r + g − b r g b + − r g b Particle statistics are particularly rich in two spatia Particle statistics are particularly rich in two sp statistics are particularly rich in two spatial dirticle statistics are particularly rich in two spatial diistics are Non-commutative particularly rich in two spatial difusion rules! (unlike anyons) nre ofbeyond the basic references). Anyonic statistics are mensions, where beyond the usual fermions and bo mensions, where beyond the usual fermions and where beyond the usual fermions and bosons ions, where beyond the usual fermions and bosons the usual fermions and bosons {e, ζ Γ = , ζ {e, , σ ζ Γ , σ = , ζ , {e, σ , σ } ζ Γ , σ = , ζ , {e, σ , σ ζ } , σ , ζ , σ , σ } , σ , σ } Γ = {e, ζ , ζ , σ , σ = {e, ζ Γ = , ζ {e, , σ ζ Γ , σ = , ζ , {e, σ , σ } ζ Γ , σ = , ζ , {e, σ , σ ζ } , σ , ζ , σ , σ } σ , σ } Γ = {e, ζ , ζ , σ , σ , σ Particle tatistics are particularly Particle statistics are particularly statistics rich are particularly in two rich are spatial particularly in two rich dispatial in two rich dispatial in two d Particle statistics are particularly rich + − r + g − b r + g − b r + g − b r g b + − r g b + − r + g − b r + g − b r + g − b r g b + − r g b ex enough to give rise to the notion of topological here exist more generally ][? there exist more generally anyons ] aforc tre more generally anyons (see [? a[?for comexist more generally anyons ](see afor comgenerally anyons (see [?anyons ] (see for] (see a[?for comwhere eyond nsions, mensions, beyond the where usual beyond the where fermions usual beyond the fermions and usual the bosons fermions and usual bosons fermions and boso an mensions, where beyond the usual ferm pilation of basic the basic references). Anyonic pilation of the basic references). Anyonic f)on the basic references). Anyonic statistics are stati of(2) the references). Anyonic statistics are basic references). Anyonic statistics are statistics  )  (b)  (c)  (d)  e  m  k  k  ck → ck  TSC twists{1, r, g, b}  {1, r, g, b} Transpositions →ζ2 possible Γ = {e, , ζ , σcharges ,σ ,σ +  −  r  g  b}  As in toric Γ codes, we ζfix = {e, , ζ− , σr , σg , σb } +one Colored Majorana operators:  σci The i-th twist is σci and i<j  kj ki if ci = ζ+ (cj ), ki kj = if ci = ζ+ (cj ), −kkijkkj i otherwise. ki kj = −ki kj otherwise.  (a)  (b) s sm TSC twists  (c)  Braiding is now more interesting  (b)  (c)  kj → kj+1 , kj+1   if cj =  −kj (e) → ikj kj+1 if cj =  −kj kj+1 otherw   if cj = cj+1 ,  −kj → ikstatistics if are cj =particularly ζ− (cj+1 ), (6) Particle rich in j → kj+1 , kj+1 j kj+1  otherwise. m j kj+1 beyond mensions,−kwhere the usual fermi  there exist more generally anyons (see [? Particle statistics are particularly rich in two spatial dipilation of the basic references). Anyon e ensions, where beyond the usual This gives the whole Cliffordfermions group!!! and bosons complex enough to(see give[?rise to athe notion ere exist more generally anyons ] for comcomputation (TQC) [? ?are? ], ation of thequantum basic references). Anyonic statistics  Conclusions Topological codes stand out for their locality TSCs only require 2-local measurements Twists reflect anyon symmetries Twist are a tool to improve topological codes With twists, via code deformation, we can implement all Clifford gates on TSCs PRL 105.030403 / arXiv:1004.1838 arXiv:1006.5260  

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