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Twisted Interferometry: the topological perspective Parsa Bondersona, Lukasz Fidkowskia, Michael Freedmana,b, Kevin Walkera 6 1 aStationQ,Microsoft Research,SantaBarbara, California93106-6105, USA 0 2 bDepartmentofMathematics, University ofCalifornia, SantaBarbara, California93106, USA c e D 9 Abstract ] h Three manifold topology isused toanalyze theeffect ofanyonic interferometers inwhich p - theprobeanyons’pathalonganarmcrossesitself,leadingtoa“twisted”orbraidedspace- t n timetrajectoryfortheprobeanyons.InthecaseofIsingnon-Abeliananyons,twistedinter- a ferometry isshowntobeabletogenerate atopologically protected π/8-phase gate,which u q cannotbegenerated fromquasiparticle braiding. [ 2 Keywords: Interferometry; Anyoniccharge measurement;Topological quantum v computation. 5 2 PACS: 03.67.Lx,03.65.Vf,03.67.Pp,05.30.Pr 2 8 0 . 1 0 1 Introduction 6 1 : v Anyonicinterferometry[1,2]isapowerfultoolforprocessingtopologicalquantum i X information [3,4,5,6,7,8,9]. Its ability to non-demolitionally measure the collec- r a tive anyonic charge of a group of (non-Abelian) anyons, without decohering their internal state, allows it to generate braid operators [10,11], generate entangling gates [12,13,14,15], and change between different qubit encodings [14,15]. Any- onicinterferometryhasbeenthefocusofmyriadexperimentalproposals[16,17,18,19,20,21,22,23,24,25,26 and efforts to physically implement them [30,31,32,33,34,35,36]. As powerful as anyonic interferometry may be, its potential capabilities have yet to be fully understood.Inthispaper,weproposeandanalyzeanovelimplementationofanyonic interferometry that we call “twisted interferometry,” which can significantly aug- mentitspotentialcapabilities. Oneoftheprimarypracticalmotivationsforstudyingtwistedinterferometryisthat it could be used with anyons of the Ising TQFT to generate “magic states,” as we willdemonstrate. This is significantbecause, if oneonly has theability to perform PreprintsubmittedtoarXiv 29January2016 braiding operations and untwisted anyonic interferometry measurements for Ising anyons, then one can only generate the Clifford group operations, which is not computationally universal and, in fact, can be efficiently simulated on a classical computer [37]. However, if one supplements these operations with magic states, then one can also generate π/8-phase gates, which results in a computationally universalgateset [38]. Theapplicationoftwistedinterferometrytogeneratingtheπ/8-phasegateforIsing anyons is the latest link in a chain of ideas [39,40,41,42], originating with the un- publishedworkofBravyiandKitaev,forgeneratingatopologically-protectedcom- putational universal gate set from the Ising TQFT by utilizing topological operations. The concept and analysis of twisted interferometry is new, but closely con- nectedtotheseideas,whichstemfromtheconceptofDehnsurgeryon3-manifolds. As we will discuss in detail, anyonic interferometry: 1) projectively measures the topological charge inside γ, and 2) decoheres the anyonic entanglement between thesubsystemsinsideandoutsidetheinterferenceloopγ[43].Bothoperationshave a3DtopologicalinterpretationinthecontextofChern-Simonstheoryor,moregen- erally, axiomatic (2+1)D topological quantum field theories (TQFTs). We learned from Witten [44] that all low energy properties of systems governed by a TQFT can be calculated in a Euclidean signature diagrammatic formalism called unitary modular tensor categories (UMTC). This suggests [40,41] that the choice of inter- ference loopγ shouldnotberestrictedto asimplespace-likeloopinaspatialslice R2 R2 time,asisthetypicaldesignforaninterferometer,butratherγ mightbe ⊂ × a general simple closed curve of space-time. Twisted interferometry explores this directionbyallowingtheprobeanyons’paththroughthearmsoftheinterferometer to beself-crossingin R2 (so γ isimmersed in mathematicalterminology).We give ageneralprocedureforanalyzinginterferometersofthiskind.Intherestrictedcase oftheIsingTQFT,wedescribeatwistedinterferometerwhichwouldbecapableof producingmagicstates. Our strategy is: 1) to start with the UMTC calculation [1,2] which lays bare the asymptotic behavior of the simplest anyonic Mach-Zehnder interferometer (and serves as a model for Fabrey-Pe´rot type interferometers in the weak tunneling limit); 2) describe this behavior in an equivalent topological language; and 3) ex- ploitthegeneral covarianceinherentinthetopologicaldescription. TheconcretecalculationusingthemachineryofUMTCsiscarriedoutinacompan- ionpaper[45],whichalsofocusesonpossiblephysicalimplementationsoftwisted interferometers. The analysis of the companion paper agrees with the topological argument presented here and both show how magic state production is achieved when specializedto theIsingtheory. 2 (cid:1371) T 2 (cid:3036)(cid:3087)(cid:3141)(cid:3141) (cid:2879)(cid:2869) A(cid:1857) (cid:1844)(cid:3002)(cid:3003) (cid:1372) B … B N 1 (cid:3036)(cid:3087)(cid:3141) (cid:1857) (cid:1844)(cid:3003)(cid:3002) T C 1 Fig. 2.1. An idealized Mach-Zehnder interferometer for an anyonic system, where T are j beamsplitters.Thetargetanyons(collectivelydenotedA)inthecentralregionshareentan- glementonlywiththeanyon(s)C outsidethisregion.AbeamofprobeanyonsB ,...,B 1 N issentthrough theinterferometer anddetected atoneofthetwopossible outputsbyD . s 2 WhatanAnyonicInterferometer Does inTwo Different Languages Werecallthebarebonesofanyonicinterferometryinageneralanyoniccontext(as developedin [1,2]; see[45]fornotationalclarificationand calculationaldetails). The target anyon A may be a composite of several quasiparticles (anyons), so it is not necessarily in an eigenstateof charge. In thesimplestcase, which we treat, the probe quasiparticles B are assumed to be uncorrelated, identical, and simple (not composites).In fact, tomake thesourcestandard and uncorrelated, the probes will beindependentlydrawnfromthevacuumtogetherwithanantiparticle(topological chargeconjugateanyon),whichisthendiscardedandmathematically“tracedout.” Wewillsimplifythediscussioninthispaperbyalsoassumingtheprobehasdefinite topologicalchargevaluesB = b,butthegeneralizationisstraightforward.Coming from the left, probe anyon B encounters first beam splitter T , and then T . The i 1 2 correspondingtransitionmatricesare: t r j j∗ T = . (2.1) j   r t  j − ∗j    Theunitaryoperatorrepresentingaprobeanyonpassingthroughtheinterferometer isgivenby U = T ΣT (2.2) 2 1 3 0 eiθIIRA−B1 Σ = . (2.3)   eiθIR 0 BA     Thiscan bewrittendiagrammaticallyas A B s t r r r r t t t U = eiθI 1 2∗ 1∗ 2∗ +eiθII 1 2 − ∗1 2 , (2.4)     Bs′ A −t1t∗2 −r1∗t∗2s,s′B A r1r2 −t∗1r2s,s′B A     where we introduce the notation of writing the directional index s of the probe quasiparticle as a subscript on its anyonic (topological) charge label, e.g. B . The s anyonicstatecomplementarytotheregionbeingprobedwillbedenotedbyC (and laterbytwo disjointsectorsC and C ). 1 2 ThepassageofasingleprobeB transformsthedensitymatrixρAC forbothsystem and environmentby 1 ρAC ρAC (s) = Tr Π VU ρB ρAC U V Π , (2.5) B s † † s 7→ Pr(s) ⊗ h (cid:16) (cid:17) i f whereTristhe“quantumtrace,” V representsbraiding,and f Pr(s) = Tr[Π VUρU V ] (2.6) s † † f is the probability of measurement outcome s. The effect of this superoperator can becomputedbyconsideringtheactionontheρAC densitymatrix’sbasiselements, whichis expresseddiagrammaticallyby a c Πs U a µ b(cid:1) f bs (2.7) a′ µ′ U† Πs a′ c′ Fortheoutcomes =(cid:1), thismaybeexpanded as 4 (Faa′cc′)−1 (f,µ,µ′)(e,α,β) (eX,α,β)h i a c a c b  e ×|t1|2|r2|2 α b β +t1r1∗r2∗t∗2ei(θI−θII) α e β a c a c  ′ a′ c ′a ′ c e +t∗1r1t2r2e−i(θI−θII)b α β +|r1|2|t2|2 b α e β  a c a c ′ ′ ′ ′a c =db (Faa′cc′)−1 (f,µ,µ′)(e,α,β)p(cid:1)aa′e,b[Faa′cc′](e,α,β)(f′,ν,ν′) f′ νν′ (2.8) (eX,α,β) h i (f′,ν,ν′) a c ′ ′ wherewehavedefined p(cid:1) = t 2 r 2M +t r r t ei(θI θII)M aa′e,b | 1| | 2| eb 1 1∗ 2∗ ∗2 − ab +t r t r e i(θI θII)M + r 2 t 2, (2.9) ∗1 1 2 2 − − a∗′b | 1| | 2| where M is the monodromy matrix M = SabS00 (with S the modular S-matrix), ab S0aS0b andθ ,θ arethenon-universalphasesassociatedwithtraversingtheinterferometer I II via the two different paths around the interferometry region. A similar calculation fors =(cid:2)gives p↑aa′e,b=|t1|2|t2|2Meb −t1r1∗r2∗t∗2ei(θI−θII)Mab t r t r e i(θI θII)M + r 2 r 2. (2.10) − ∗1 1 2 2 − − a∗′b | 1| | 2| Thus,wehavethesingleprobemeasurementprobabilities Pr(s) = ρAC ps , (2.11) (a,c;f,µ),(a,c;f,µ) aa0,B a,c,f,µ X and post-measurementstate(foroutcomes) ρAC ρAC (s)= (a,c;f,µ),(a′,c′;f,µ′) Fa,c −1 a,a′,cX,c′,f,µ,µ′ (dfdf′)1/2 (cid:20)(cid:16) a′,c′(cid:17) (cid:21)(f,µ,µ′)(e,α,β) (e,α,β),(f′,ν,ν′) ps aa′e,B Fa,c a,c;f ,ν a,c;f ,ν . (2.12) ×Pr(s) a′,c′ (e,α,β)(f′,ν,ν′)| ′ ih ′ ′ ′ ′| h i 5 The next step (which we sketch very lightly here) is to compute probabilities and theeffect forastreamofN identicalprobeanyonsB, on ρAC. Theresultsare: Pr(s ,...,s ) = ρAC ps1 ...psN , (2.13) 1 N (a,c;f,µ),(a,c;f,µ) aa0,B aa0,B a,c,f,µ X ρAC ρAC (s ,...,s ) = (a,c;f,µ),(a′,c′;f,µ′) Fa,c −1 1 N a,a′,cX,c′,f,µ,µ′ (dfdf′)1/2 (cid:20)(cid:16) a′,c′(cid:17) (cid:21)(f,µ,µ′)(e,α,β) (e,α,β),(f′,ν,ν′) ps1 ...psN aa′e,B aa′e,B Fa,c a,c;f ,ν a,c;f ,ν . (2.14) × Pr(s ,...,s ) a′,c′ (e,α,β)(f′,ν,ν′)| ′ ih ′ ′ ′ ′| 1 N h i Itisclearthatthespecificorderofthemeasurementoutcomesisnotimportant,but only the total number of outcomes of each type matters, and that keeping track of onlythetotalnumbersleadsto abinomialdistribution. Forgeneric choices of interferometricparameters: t ,r ,θ , and θ , thesebinomial j j I II distributions will concentrate exponentially fast at distinct transmission probabilities associated with the equivalence classes of charge types a where a a if and ′ ≡ only if Ma,b = Ma′,b. In the simplest cases, there is a natural choice for the probe B where every a is distinguished (e.g. for Ising and Fibonacci anyons one selects b = σ andb = τ, respectively),andhencethe“equivalenceclasses”aresingletons. In general, theprobabilityofobservingn(out ofN)probes inthe detectoris: → N! Prκ (n)= Pr (κ) pn(1 p )N n, (2.15) N A n!(N n)! κ − κ − Xκ − Pr (κ)= ρAC , (2.16) A (a,c;f,µ),(a,c;f,µ) a∈CXκ,c,f,µ where κ indexes the equivalence classes w.r.t. probe b. The fraction r = n/N κ C of probes measured in the s = detector goes to r = p with probabilityPr (κ), κ A → andthetargetanyondensitymatrixwillgenericallycollapseontothecorresponding “fixed states.” The asymptotic operation N of a generically tuned anyonic interferometer → ∞ convergestoafixedstateofchargesectorκwithprobabilityPr (κ)and:1)projects A the anyonic state onto the subspace where the A anyons have collective anyonic chargein ,and2)decoheresallanyonicentanglementbetweensubsystemAand κ C C that the probes can detect. The sector κ may be a single charge or a collection ofcharges withidenticalmonodromyelementswith theprobes,i.e. Ma,B = Ma′,B fora,a .TheanyonicentanglementbetweenAandC isdescribedintheform ′ κ ∈ C of anyonic charge lines connecting these subsystems, i.e. the charge lines labeled by charge e in the preceding analysis, where the contribution of a diagram to the density matrix will be removed if M = 1. Convergence to such a fixed state is e,B 6 6 C (cid:1371) T 2 2 A (cid:1372) B … B N 1 T C 1 1 Fig. 2.2. An idealized Mach-Zehnder interferometer where the anyons C entangled with thetargetanyonsAareseparated intotworegionsC andC . 1 2 based on Gaussian statistics, therefore exponentially precise as a function of the numberN ofprobeparticles. In thesimplestcase, Ma,b = Ma′,b a = a′ and theindistinguishableequivalence ⇒ classesC = a aresingletons,i.e.alltopologicalchargesaredistinguished.The κa { } correspondingfixedstatedensitymatrixis: Pr (c a) Pr (c a) ρAC = A | I = A | a,c;f ,ν a,c;f ,ν , (2.17) κa d d ac d d | ′ ih ′ | c a c c,f′,ν a c X X where ρA (a,c;f,µ)(a,c;f,µ) f,µ Pr (c a) = . (2.18) A | P ρA (a,c;f,µ)(a,c;f,µ) c,f,µ P (The formulae for the general case can be found in [1,2].) From this point on, we focus onlyonthesecases wheretheprobedistinguishesalltopologicalcharges. This is a convenient place to note a modest generalization, where the complemen- tarychargeC isdividedintotworegionsseparatedbytheinterferometer,whichwe similarly denote as C and C , respectively. In some experimental setups — e.g. 1 2 a Fabrey-Pe´rot interferometer on a quantum Hall bar — each arm of the interferometer individually will separate the region with charge A from a complementary region with respective charges C and C , which could both be nontrivial. This 1 2 situation is depicted for the idealized Mach-Zehnder interferometer in Fig. 2.2. In this circumstance, all charge lines from A to C and from A to C are (separately) 1 2 decohered iftheycan bedetected bytheprobes B. 7 a c c a c 1 2 α 2 α α 3 2 α α α α 4 1 4 1 e e e 2 1 α 3 a′ c′ c′1 a′ c′2 (a) (b) Fig. 2.3. (a) For a single region of complementary anyons C, we show the four positions for the probe loops corresponding to the four terms of Eq. (2.8). (b) For two regions of complementarychargeC andC ,thefourpositionsofprobeloopsareshownonthemore 1 2 complicated the target system (with complementary anyons) density matrix components. (The 4-valent vertex is understood to be resolved into appropriate trivalent vertices.) α j denotestheweightwithwhichthecorresponding probeloopconfiguration entersthemea- surementsuperoperator. In Fig. 2.3, we compare the diagrammatic terms that arise for a single C region formulation to when there are two regions C and C . For probe b and measure- 1 2 ment outcome s =(cid:1), the four probe loop configurations enter the measurement superoperatorwithweights α(cid:1)= t 2 r 2, (2.19) 1 | 1| | 2| α(cid:1)=t r r t ei(θI θII), (2.20) 2 1 1∗ 2∗ ∗2 − α(cid:1)=t r t r e i(θI θII), (2.21) 3 ∗1 1 2 2 − − α(cid:1)= r 2 t 2, (2.22) 4 | 1| | 2| as inEq. (2.8). Fors =(cid:2),theseare α(cid:2)= t 2 t 2, (2.23) 1 | 1| | 2| α(cid:2)= t r r t ei(θI θII), (2.24) 2 − 1 1∗ 2∗ ∗2 − α(cid:2)= t r t r e i(θI θII), (2.25) 3 − ∗1 1 2 2 − − α(cid:2)= r 2 r 2. (2.26) 4 | 1| | 2| GivenN uncorrelatedidenticalprobeanyons,thereare4N configurationsofprobe loops, each probe choosing from the four positions, with the single probe weights (depending on a given probe’s measurement outcome) being multiplied together for the overall superoperator. For the two probe loop positions which cross in Fig. 2.3(b), repeated copies will nest according to the pattern of later probe loops having larger radius. We will see shortly that the detail of the nesting patterns are irrelevantinthelargeN limit. 8 Accordingtothecalculationjustsummarized,theneteffectofrunningtheinterfer- ometer on the target system with density matrix ρAC, up to corrections that decay exponentiallyinN, isthat thesuperpositionofthese4N configurations resultsina measurement of the collectivecharge of anyons A onto charge valuea, with probability Pr (a) = Tr ρACΠA , (2.27) AC a h i and post-measurementdensitymatrix f C A C 2 1 1 ρAaC = Pr (a) ω0 ωa ρAC ωa ω0 . (2.28) AC C A C 2′ ′ 1′ (All topological charge lines drawn here have zero framing, i.e. there are no twists intheframe.)Theω -loops a = = S S (2.29) 0a a∗x ωa ωa¯ Xx x havetheeffectofprojectingallchargelinespassingthroughtheloopontocollective charge a. Thus, the ω -loops effectively cut charge lines. This allows the ω -loops 0 a to be moved to encircle only the A and A lines, i.e. one can perform a handle ′ slide of the loop around the ω -loops (see Section 3.1). Thus, the ω -loops effect 0 a projection of anyons A into collective charge sector a. When there is only one region of complementary anyons C, e.g. if there are no C anyons, then the action 2 of the ω -loop between A and C is trivial. Notice that the ω-loops here occur in 0 2 preciselythesamepositionsas thefourpossibleprobeloopconfigurations. Having depicted the effect of interferometry in terms of ω-loops, we make a ge- ometric observation for later use: the effects of interferometry are localized to a certain quasi-1D region of space-time surrounding the ω-loops called a “handle body.”TheseareindicatedinFig.2.4astheregionsH andH forthesingleregion ′ C and two regionC and C configuration of complementary anyons.The handle- 1 2 bodies H and H model the complementary regions surrounding the ρAC density ′ matrix operator. This enables us to make calculations for twisted interferometry simplyby computingoperatorswithintransformedcoordinates. 9 H′ ωa H ω a β γ ω a γ ω0 ω0 ω0 γ¯ β¯ γ¯ ω a (a) (b) Fig. 2.4. (a) Genus 2 handle body H and (b) genus 3 handle body H , within which the ′ effectofinterferometry islocalized inEq.(2.28).Somecurvesin∂H and∂H arelabeled ′ forlaterreference. 3 TopologicalExplanations The goal of this section is to explain the topological nature of interferometry. In Section3.1,wefirstreviewsomepuretopologybackgroundon3-manifoldsurgery and the handle slide property. In Section 3.2, we apply this machinery to interferometry, with the basic idea being that in the limit of large N, the exact partition function,givenby 4N terms withprobeanyonWilsonloops,can effectivelybede- scribed by a small number of Dehn surgeries. Although this abstract topological approachmayatfirstseemlikeoverkill,itprovesitsutilitywhenwetrytogeneral- izetothecaseoftwistedinterferometry,whichisintroducedinSection3.3.Indeed, as shownin Section 3.4, twistinghas anatural descriptionintheeffectivetopolog- ical language: to compute the partition function in the twisted case, all we have to do is modify the gluing of a certain handle body by some twists. Section 3.5, although not necessary in the logical flow of the paper, develops a stand-alone, purely topological perspective on interferometry. Finally, in Section 3.6, we apply all this machinery to the case of the Ising UMTC, and describe the simplifications thatarise. 3.1 SurgeryandtheHandleSlideProperty “Handles”areacombinatorialtoolforassemblingsmoothd-manifoldswithbound- ary out of littlepieces, which are individuallycopies of d-balls. Our main focus is d = 4, since we will manipulate within a (2 + 1)D TQFT using a representation where the 3D space-time is the boundary of a 4D bulk. Note, however, that the handlebodiesdrawn inFig. 2.4are 3D,being subsetsofthespace-timeitself. Let Bd denote the unit ball in Rd. There are d + 1 types of d-dimensional han- 10