Topological Quantum Computation, Yang-Baxter Operators and. Operators and Modular Categories

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1 Topological Quantum Computation, Yang-Baxter Operators and Modular Categories March 2013, Cordoba, Argentina Supported by USA NSF grant DMS Joint work with C. Galindo, P. Bruillard, R. Ng, S.-M. Hong, Z. Wang

2 What is a Quantum Computer? From [Freedman-Kitaev-Larsen-Wang 03]: Definition Quantum Computation is any computational model based upon the theoretical ability to manufacture, manipulate and measure quantum states.

3 Mathematics of Quantum Mechanics Basic Principles Superposition: a state is a vector in a Hilbert space ψ H Entanglement: Composite system state space is H 1 H 2 Schrödinger: Evolution of the system is unitary U U(H) Observables O: Spin, position, energy, etc. a Hermitian operator H O. Wave-function collapse: Measuring O on ψ = i a i e i gives H O -eigenstate e i with probability a i 2.

4 Quantum Circuit Model Fix d Z and let V = C d. Definition The n-qudit state space is the n-fold tensor product: M n = V V V. A quantum gate set is a collection S = {U i } of unitary operators U i U(M ni ) (n i -local) usually n i 4. Definition A quantum circuit for U on S = {U i } is: G 1,..., G m U(M n ) where G j = I a V U i I b V G 1 G 2 G m U < ɛ. and

5 Topological Phases of Matter (anyons) Definition Topological Quantum Computation (TQC) is a computational model built upon systems of anyons (topological phases). Fractional Quantum Hall Liquid electrons/cm 2 T 9 mk quasi-particles B z 10 Tesla

6 Anyons? In R 3, bosons or fermions: ψ(z 1, z 2 ) = ±ψ(z 2, z 1 ) Particle exchange reps. of symmetric group S n In R 2 : anyons: ψ(z 1, z 2 ) = e iθ ψ(z 2, z 1 ) Particle exchange reps. of braid group B n =

7 The Braid Group Definition B n has generators σ i, i = 1,..., n 1 satisfying: (R1) σ i σ i+1 σ i = σ i+1 σ i σ i+1 (R2) σ i σ j = σ j σ i if i j > 1 1 i i+1 n σ i

8 Topological Model (non-adaptive) Computation Physics output measure (fusion) apply gates braid anyons initialize create anyons vacuum

9 Mathematical Model for Anyons Definition (Nayak, et al 08) a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory (TQFT). Fact Most (known, useful) (2 + 1)-TQFTs come from the Reshetikhin-Turaev construction via modular categories.

10 Modular Categories Definition A modular category C is a balanced, braided, fusion category with a non-degenerate braiding. Examples Rep(DG): double of a finite group G. C(g, l) obtained from U q g at q = e πi/l.

11 Algebraic Data Remark A fusion category C has simple iso-classes {X 0 = 1,..., X k 1 } (rank k), fusion rules X i X j = m Nm i,j X m and duality X X. Maximal eigenvalue of (N i ) m,j = N m i,j is denoted FPdim(X i ). FPdim(C) = i FPdim(X i) 2 A braiding c X,Y : X Y Y X makes commutative. Balancing θ X Aut C (X ): and c are compatible. modularity S X,Y = Tr C (c X,Y c Y,X ) invertible, gives representation of SL(2, Z) via S and T X,Y = δ X,Y θ X.

12 Topological Model Remarks state spaces of type X C: H i n := Hom(X i, X n ) Physically, gates are particle exchanges acting on H n := i Hi n. Mathematically, gates are {ϕ X n (σ i )} where ϕ X n : B n i U(H i n). i i+1 i

13 Example: C(sl 2, l) Example C(sl 2, l) has simple objects {X 0,..., X l 2 } and: S i,j = sin( (i+1)(j+1)π l ) sin( π l ) X 1 X k = X k 1 X k+1 θ j = e πi(j2 +2j)/(2l) FPdim(X k ) = sin( (k+1)π l ) sin( π l )

14 Problem 1 Problem Classify modular categories by rank. Feasible? Conjecture (Wang 2003) For each k N, there are finitely many modular categories C with rank(c) = k. Physically: finitely many systems with exactly k indistinguishable, indecomposable particle types (anyons).

15 Towards Wang s Conjecture Definition C is weakly integral if FPdim(C) Z, and integral if FPdim(X i ) Z and pointed if FPdim(X i ) = 1. Examples Rep(DG) is integral, C(sl 2, l) is weakly integral for l {3, 4, 6}. Theorem (Etingof,Nikshych,Ostrik 2005) There are finitely many weakly integral fusion categories of rank k. Remark Integral modular categories of rank 7 are pointed, weakly integral modular categories of rank 6 are classified, pointed modular categories are classified.

16 Theorem (Bruillard,Ng,R.,Wang) There are finitely many (unitary) modular categories of rank k. Proof Sketch. Order of T = Diag(θ 0,..., θ k 1 ) M 1 (k). Cauchy theorem for (unitary) modular categories: ord(t ) and FPdim(C) have the same prime divisors (in Z[e 2πi/ord(T ) ]). S-unit equation (number theory) implies: FPdim(C) M 2 (k) 0 Ni,j k FPdim(C), so finitely many Nk i,j. Ocneanu rigidity implies finiteness. Remark Classification still hard: algorithm quadruply exponential!

17 Low Rank Classification Theorem ((R.,Stong,Wang),(Bruillard,Ng,R.,Wang)) Up to K 0 (C) (fusion rules): Rank 2: C(sl 2, 3), 1 2 C(sl 2, 5) Rank 3: Rep(Z 3 ), C(sl 2, 4), 1 2 C(sl 2, 7) Rank 4: Products, or Rep(Z 4 ), 1 2 C(sl 2, 9) Rank 5: Rep(Z 5 ), 1 2 C(sl 2, 11), 1 3 C(sl 3, 7), C(sl 2, 6).

18 Classification Tools For modular categories of rank k with ord(t ) = N: Q(θ i ) Z d 2 Z N Q(S i,j ) H<S k Q Consequences [Q(S i,j ) : Q] = p > 3 prime implies 2p + 1 is prime. (0 1)(2 k 1) H 0 is H-fixed if and only if C is integral.

19 Problem 2 Fact Topological Models are not typically local. Question H n = i Hom(X i, X n ) V f (n) ϕ X n (σ i ) smeared across all of H n. When can we uniformly localize (ϕ X n, H n )? Why? Simulate topological quantum computers directly on quantum circuits.

20 Local B n representations: Yang-Baxter eqn. Definition (R, V ) is a braided vector space if R Aut(V V ) satisfies (R I V )(I V R)(R I V ) = (I V R)(R I V )(I V R) Induces a sequence of local B n -reps (ρ R, V n ) by ρ R (σ i ) = I i 1 V R I n i 1 V v 1 v i v i+1 v n ρ R (σ i ) v 1 R(v i v i+1 ) v n

21 Square Peg, Round Hole? Definition (R,Wang) A localization of a sequence of B n -reps. (ρ n, V n ) is a braided vector space (R, W ) and injective algebra maps τ n : Cρ n (B n ) End(W n ) such that the following diagram commutes: CB n ρ n Cρ n (B n ) ρ R τ n End(W n ) Quasi-localization also defined: see [Galindo,Hong,R. 2013].

22 Example C(sl 2, 4) Let R = Theorem (Franko,R,Wang 2006) (R, C 2 ) localizes (ρ X n, H n ) for X = X 1 C(sl 2, 4) Remark Notice: X is not a vector space! (FPdim(X ) = 2)

23 Some Answers Theorem H braided Hopf algebra: X Rep(H), B n -reps. (ϕ X n, H n ) always localizable. Remark B n acts on both X n and H n := i Hom(X i, X n ). The first is local the second is localizable. Theorem (R,Wang) For X C(sl 2, l) (ϕ X n, H n ) localizable if and only if FPdim(X ) 2 Z (i.e. C[X ] weakly integral).

24 Localization Conjecture Conjecture (ϕ X n, H n ) is (quasi-)localizable for all X C if and only if C is Weakly Integral. Theorem (R,Wang 2012) Suppose that each H i n is irreducible as a B n -rep. then Localizable Weakly Integral. proof: Spectral Graph Theory Remark Observe that ϕ X n (σ i ) has finite order: eigenvalues are roots of unity.

25 Unitary BVS Conjecture Question How do local braid group representations behave? Conjecture If (R, V ) is a unitary, finite order BVS then ρ R (B n ) <.

26 Reduction Theorem Suppose (R, V ) is unitary, finite order BVS. Then if ρ R (B n ) is finite modulo its center then ρ R (B n ) < Proof. restrict to irred. subrep. (ρ R i, W i ) (ρ R, V n ). ρ R i (σ j ) has finite order, so for z = xid Wi Z(ρ R i (B n )), det(z) = x dim(w i ) is a root of unity. Thus z has finite order so Z(ρ R (B n )) is f.g. torsion, hence finite.

27 Classification Approach 1992 [Hietarinta] classified BVSs with dim(v ) = [Dye] classified BVSs with dim(v ) = 2 and unitary 2006 [R, Franko, Wang] Conjecture verified for BVSs dim(v ) = 2 and unitary. Generally hopeless: solve dim(v ) 6 equations in dim(v ) 4 variables! dim(v ) 3 open.

28 Group Type BVSs Definition (R, V ) is of group type if there exist g i GL(V ) and a basis {x i } such that R(x i x j ) = g i (x j ) x i. Remark Group-type BVSs are important to Andruskiewitsch-Schneider classification program for pointed Hopf algebras. Theorem Conjecture true for group type BVSs if G := g 1,..., g n is finite. Proof. V CG CG YD with braiding c V,V = R. CG CG YD = Rep(DG) so that [Etingof,R.,Witherpoon] implies ρ R (B n ) <.

29 Theorem (R.) If (R, V ) is a BVS then G = g 1,..., g n is finite modulo Z(G). Proof Idea. The YB equation is equivalent to g j,k i g i g j = g j,k i g k g i.g acts on {g 1,..., g n } by conjugation. Kernel is Z(G). Question If each g i has finite order, or if G is abelian, conjecture is true. Does the above imply ρ R (B n ) is finite modulo it center?

30 Full Circle... Definition A braided fusion category has Property F if ϕ X (B n ) < for all X. Conjecture Let C be a braided fusion category. Then the following are equivalent: 1 C has Property F 2 (ϕ X, H n ) is quasi-localizable for all X 3 C is weakly integral.

31 Thank you!