Acyclic anyon models, thermal anyon error corrections, and braiding universality
AACYCLIC ANYON MODELS, THERMAL ANYON ERRORCORRECTIONS, AND BRAIDING UNIVERSALITY
C´ESAR GALINDO, ERIC ROWELL, AND ZHENGHAN WANG
Abstract.
Acyclic anyon models are non-abelian anyon models for which thermalanyon errors can be corrected. In this note, we characterize acyclic anyon modelsand raise the question if the restriction to acyclic anyon models is a deficiency of thecurrent protocol or could it be intrinsically related to the computational power ofnon-abelian anyons. We also obtain general results on acyclic anyon models and findnew acyclic anyon models such as SO (8) and untwisted Dijkgraaf-Witten theories ofnilpotent finite groups. Introduction
In topological quantum computing (TQC), information is encoded in the ground statemanifolds of topological phases of matter which are error correction codes. Therefore,TQC is intrinsically fault-tolerant against local errors. But at any finite temperature
T >
0, thermal anyon pairs created from the vacuum due to thermal fluctuations candiffuse and braid with computational anyons to cause errors, the so-called thermalanyon errors. In practice, thermal anyon creations are suppressed by the energy gap ∆and low temperature T as αe − ∆ T for some positive constant α , so it might not pose aserious challenge. But if the suppression by gap and temperature is not enough, thenthermal anyon errors could become a serious issue for long quantum computation. In[DP17], the authors found an error correction scheme for acyclic anyon models (callednon-cyclic in [DP17]). In this paper, we characterize acyclic anyon models as anyonmodels with nilpotent fusion rules. We obtain several general results on acyclic anyonmodels and find many more acyclic anyon models such as SO (8) , which has Property F . Date : January 16, 2018.2000
Mathematics Subject Classification. T (INVERSION TEMPORAL) EN CATEGOR´IAS DE FUSI ´ON Y MODU-LARES”, E.R. was partially funded by NSF grant DMS-1664359, and Z.W. was partially funded byNSF grant DMS-1411212 and FRG-1664351. a r X i v : . [ m a t h . QA ] J a n GALINDO, ROWELL, AND WANG
Our characterization of acyclic anyon models raise the question if the restriction toacyclic anyon models is a deficiency in the current protocol or could it be intrinsi-cally related to the computational power of non-abelian anyons. A triality exists forthe computational power of non-abelian anyons as illustrated by the anyon models SU (2) k , k = 2 , ,
4. The type of anyons in SU (2) k is labeled by the truncated angularmomenta in { , / , . . . , k/ } and let s be the spin=1 / k = 2, s isessentially the Ising anyon σ , not only it is not braiding universal, but also all braidingcircuits can be efficiently simulated by a Turing machine. Moreover, it is believed thatall measurements of total charges can also be efficiently simulated classically. When k = 3, s is the Fibonacci anyon, which is braiding universal [FLW02]. When k = 4, s is a metaplectic anyon which is not braiding universal. But supplemented by a totalcharge measurement, a universal quantum computing model can be designed based onthe metaplectic anyon s [CW15]. While SU (2) is acyclic, neither SU (2) nor SU (2) is. Since acylic anyon models are weakly integral (proved below), they should not bebraiding universal as the property F conjecture suggests [NR11]. Therefore, it wouldbe interesting to know if any acylic model can be made universal when supplementedwith total charge measurements. If not, then whether or not the protocols in [DP17]can be generalized to go beyond acyclic anyon models.2. Preliminaries
An anyon model is mathematically a unitary modular tensor category—a very dif-ficult and complicated structure [RW17]. But the fusion rule of an anyon model iscompletely elementary. Our main result is a theorem about fusion rules, so we startwith the basics of fusion rules to make the characterization of acyclic anyon modelself-contained.2.1.
Fusion Rules. A fusion rule (A,N) based on the finite set A is a collection ofnon-negative integers { N kij } as below, where the elements of A will be called anyontypes or particle types or topological charges. The elements in A will be denoted by x , x , x , . . . . A fusion rule is really the pair ( A, N ), but in the following we sometimessimply refer to the set A or the set of integers { N kij } as the fusion rule when no confusionwould arise.For every particle type x i there exists a unique dual or anti-particle type, that wedenote by x i = x i . There is a trivial or “vacuum” particle type denoted by 1.The fusion rules can be conveniently organized into formal fusion product and sumof particle types (mathematically such formal product and sum can be made into oper-ations of a fusion algebra where particle types are bases elements of the fusion algebra): CYCLIC ANYON MODEL 3 x i x j = (cid:88) k N ki,j x k where N ki,j ∈ Z ≥ . The fusion rules obey the following relations(a) Associativity: ( x i x j ) x k = x i ( x j x k ),(b) The vacuum is the identity for the fusion product, x i x i = 1 x i , (c) The anti-particle type x i (cid:55)→ x i = x i defines an involution, that is,1 = 1 , x i = x i , x j x i = x i x j , where x i x j := (cid:88) k N ki,j x k , (d) The fusion of x i with its antiparticle x i contains the vacuum with multiplicityone, that is N i,i = 1 . A fusion rule is called abelian (or pointed) if (cid:88) k N ki,j = 1for every x i and x j . If A is an abelian fusion rule, then the fusion product defines agroup structure on A and conversely every group defines an abelian fusion rule.2.2. Nilpotent fusion rules.
Let (
A, N ) be a fusion rule on the set A . A sub-fusionrule of ( A, N ) is a subset B ⊂ A such that(a) 1 ∈ B ,(b) x i ∈ B if and only if x i ∈ B ,(c) if x i , x j ∈ B , then N ki,j > x k ∈ B .The rank of the fusion rule A is | A | , the cardinality of the set A . Definition 2.1. [GN08] Let A ad be the minimal sub-fusion rule of A with the propertythat x i x i belongs to A ad for all x i ∈ A ; that is, A ad consists of all particle types of A contained in x i x i for all x i ∈ A . Definition 2.2. [GN08] The descending central series of A is the series of sub-fusionrules · · · A ( n +1) ⊆ A ( n ) ⊆ · · · ⊆ A (1) ⊆ A (0) = A, defined recursively as A ( n +1) = A ( n ) ad , for all n ≥ GALINDO, ROWELL, AND WANG
Definition 2.3. [GN08] A fusion rule is called nilpotent , if there exists an n ∈ N suchthat A ( n ) has rank one. The smallest number n for which this happens is called the nilpotency class of A .3. Acyclic fusion rules are nilpotent
In this section, we prove our main result.3.1.
Acyclic fusion rules.Definition 3.1. [DP17] A fusion rule A is called acyclic if for any value of n ∈ N andfor any sequence ( x i = x i n +1 , x i n , . . . , x i , x i , x i )with x i (cid:54) = , we have that n (cid:89) k =1 N i k i k +1 ,i k +1 = 0 . To any fusion rule we may associate its adjoint graph defined as follows ([DP17]):the vertices are pairs X i := ( x i , x i ) and a directed edge is drawn from X i (cid:54) = ( , ) to X j if N ji,i (cid:54) = 0. Notice that this is unambiguous since N ji,i = N ji,i . An example is foundin Figure 1. Now we can alternatively say that a fusion rule is acyclic if its adjointgraph contains no directed cycles. The adjoint graph found in Figure 1 corresponds to SO (8) , an integral modular category of dimension 32 and rank 11: the explicit fusionrules are found in [BGPR17]. Notice that there are no directed cycles in the adjointgraph of SO (8) so its fusion rule is acyclic. Note also that the direct product of twoacyclic fusion rules is acyclic as well. Lemma 3.2.
Let A be finite acyclic fusion rule with | A | > . Then the rank of A ad isstrictly smaller than the rank of A .Proof. Assume that A is acyclic and A ad = A .For each n ∈ N we will define a sequence of bases elements(3.1) ( x i n , . . . , x i , x i )such that(a) N i k i k +1 ,i k +1 > N i k +1 i k ,i k = 0 for all k < n .(b) x i k (cid:54) = 1 for all k .Since A has rank bigger than one, there is an x i (cid:54) = 1. Using that A ad = A , we havethat there is x i such that N i i ,i >
0. Now, since A is acyclic, using the sequence( x i , x i , x i ), we have that N i i ,i = 0 . In particular, N i i ,i = 0 implies x i (cid:54) = 1. Using CYCLIC ANYON MODEL 5 X b b b b Y Y V V W W Figure 1. SO (8) adjoint graph: b , b , b b are bosons and the remain-ing objects have dimension 2. All objects are self-dual.the same argument, we can construct for each n ∈ N a sequence ( x i n , . . . , x i ) thatsatisfies (a) and (b).Let us see that the elements in the sequence (3.1) are pairwise distinct. For n = 2,we have that N i i ,i (cid:54) = N i i ,i , then x i (cid:54) = x i . Assume that any sequence of n − x i n , . . . , x i ) and( x i n − , . . . , x i ) are pairwise distinct. If x i = x i n , since A is acyclic, we have that n (cid:89) k =1 N i k i k +1 ,i k +1 = 0 . But by construction, N i k i k +1 ,i k +1 >
0, hence we have a contradiction. In conclusion, theelements in the sequence ( x i n , . . . , x i ) are pairwise distinct.Finally, since the rank of A is a finite number, and we can construct an arbitrarylarge sequence of pairwise distinct basic elements, we obtain a contradiction. Thus if GALINDO, ROWELL, AND WANG A is a nontrivial acyclic fusion rules, the rank of A ad is strictly smaller than the rankof A . (cid:3) Theorem 3.3.
Let A be a fusion rule. Then A is acyclic if and only if A is nilpotent.Proof. Clearly any sub-fusion rules of acyclic fusion rules is acyclic.Assume that A is acyclic. Using Lemma 3.2 we obtain that in the sequence · · · A ( n +1) ⊆ A ( n ) ⊆ · · · ⊆ A (1) ⊆ A (0) = A, the rank of A ( n +1) is strictly smaller than the rank of A ( n ) if the rank of A ( n ) is biggerthan one. Since the rank of A is finite, there is m ∈ N such that the rank of A ( m ) isone, that is, R is nilpotent.Assume that A is nilpotent. We will use induction on the nilpotency class. If A hasnilpotency class one, then A is abelian (pointed in mathematical terminology) and thusacyclic. Assume that that A is nilpotent with A ad (cid:54) = A . By induction hypothesis A ad is acyclic. Let ( x i = x i n +1 , x i n , . . . , x i , x i )be a sequence of basic elements with x i (cid:54) = 1. If N i k i k +1 ,i k +1 > k , then x i k ∈ A ad for all k and n (cid:89) k =1 N i k i k +1 ,i k +1 > , a contradiction since A ad is acyclic. (cid:3) Corollary 3.4. (1) If a gauging B × ,GG of a modular category B by a finite group G has acyclic fusion rules then B has acyclic fusion rules and G is nilpotent.(2) The untwisted Dijkgraaf-Witten theory Z (Vec G ) has acyclic fusion rules if andonly if G is a nilpotent group.Proof. (1) If B × ,GG is nilpotent any fusion subcategory is also nilpotent. Since Rep( G ) ⊂B G ⊂ B × ,GG , we have that Rep( G ) and B G are nilpotent. The forgetful functor B G → B is suryective, then by [GN08, Proposition 4.6](2) An untwisted Dijkgraaf-Witten theory of a finite group G is exactly Z (Rep( G ))the Drinfeld center of Rep( G ). Thus, by [GN08, Theorem 6.11] Z (Rep( G )) is nilpotentif and only if G is nilpotent. (cid:3) A braided fusion category C has property F [NR11] if, for every simple object X ,the braid group representation associated with X has finite image. Conjecturally, theclass of braided fusion categories with property F coincides with the class of braidedweakly integral fusion categories. It follows from [NR11] that the acyclic braided fusion CYCLIC ANYON MODEL 7 category SO (8) mentioned above has property F . Although we do not know if allacyclic braided fusion categories have property F , some partial results in this directionare as follows. Corollary 3.5. (1) A fusion category with acyclic fusion rules is weakly group-theoretical. In particular it is weakly integral.(2) An integral braided fusion category with acyclic fusion rules is group-theoreticaland hence property F .(3) A modular tensor category B has acyclic fusion rules if and only if B is theDeligne product of modular categories of prime powers.Proof. In light of Theorem 3.3, the statements follow from the results of [DGNO07,ERW08, ENO11]. (cid:3)
This strongly suggests that anyon models with acyclic fusion rules are never braidinguniversal alone.
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GALINDO, ROWELL, AND WANG
Departamento de Matem´aticas, Universidad de los Andes, Bogot´a, Colombia
E-mail address : [email protected] Department of Mathematics, Texas A&M University, College Station, TX
E-mail address : [email protected] Microsoft Research Station Q, and Department of Mathematics, University of Cal-ifornia, Santa Barbara, CA
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