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Properties of Modular Categories and their Computation Consequences

Properties of Modular Categories and their Computation Consequences. Eric C. Rowell, Texas A&M U. UT Tyler, 21 Sept. 2007. A Few Collaborators. Publications/Preprints. [Franko,ER,Wang] JKTR 15 , no. 4, 2006 [Larsen,ER,Wang] IMRN 2005 , no. 64 [ER] Contemp. Math. 413 , 2006

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Properties of Modular Categories and their Computation Consequences

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  1. Properties of Modular Categoriesand theirComputation Consequences Eric C. Rowell, Texas A&M U. UT Tyler, 21 Sept. 2007

  2. A Few Collaborators

  3. Publications/Preprints • [Franko,ER,Wang] JKTR15, no. 4, 2006 • [Larsen,ER,Wang] IMRN2005, no. 64 • [ER] Contemp. Math. 413, 2006 • [Larsen, ER] MP Camb. Phil. Soc. • [ER] Math. Z 250, no. 4, 2005 • [Etingof,ER,Witherspoon] preprint • [Zhang,ER,et al] preprint

  4. Motivation (Turaev) Modular Categories 3-D TQFT (Freedman) definition Top. Quantum Computer (Kitaev) Top. States (anyons)

  5. What is a Topological Phase? [Das Sarma, Freedman, Nayak, Simon, Stern] “…a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory…” Working definition…

  6. Topological States: FQHE 1011 electrons/cm2 particle exchange 9 mK fusion defects=quasi-particles 10 Tesla

  7. Topological Computation Computation Physics output measure apply operators braid create particles initialize

  8. MC Toy Model: Rep(G) • Irreps: {V1=C, V2,…,Vk} • Sums VW, tens. prod. VW, duals W* • Semisimple: each W=imiVi • Rep: SnEndG(V n)

  9. Modular Categories deform axioms group G Rep(G) Modular Category Snaction (Schur-Weyl) Bnaction (braiding)

  10. i+1 i 1 n Braid Group Bn“Quantum Sn” Generated by: bi= Multiplication is by concatenation: =

  11. Modular Category • Simple objects {X0=C,X1,…XM-1} + Rep(G) properties • Rep. Bn End(Xn) (braid group action) • Non-degeneracy: S-matrix invertible

  12. Uses of Modular Categories • Link, knot and 3-manifold invariants • Representations of mapping class groups • Study of (special) Hopf algebras • “Symmetries” of topological states of matter. (analogy: 3D crystals and space groups)

  13. Partial Dictionary

  14. In Pictures Simple objects Xi Quasi-particles Unit object X0 Vacuum Particle exchange Braiding Create X0Xi Xi*

  15. Two Hopf Algebra Constructions quantum group semisimplify g Uqg Rep(Uqg) F(g,q,L) Lie algebra qL=-1 twisted quantum double GDGRep(DG) finite group Finite dimensional quasi-Hopf algebra

  16. Other Constructions • Direct Products of Modular Categories • Doubles of Spherical Categories • Minimal Models, RCFT, VOAs, affine Kac-Moody, Temperley-Lieb, and von Neumann algebras…

  17. Groethendieck Semiring • Assume self-dual: X=X*. For a MC D: Xi Xj= k NijkXk (fusion rules) • SemiringGr(D):=(Ob(D),,) • Encoded in matrices (Ni)jk = Nijk

  18. Generalized Ocneanu Rigidity Theorem: (see [Etingof, Nikshych, Ostrik]) For fixed fusion rules { Nijk } there are finitely many inequivalent modular categories with these fusion rules.

  19. Graphs of Fusion Rules • Simple XimultigraphGi : Vertices labeled by 0,…,M-1 Nijk edges j k

  20. 0 1 2 3 Example: F(g2,q,10) Rank 4 MC with fusion rules: N111=N113=N123=N222=N233=N333=1; N112=N122= N223=0 G1: Tensor Decomposable, 2 copies of Fibbonaci! G2: 0 2 1 3 G3: 0 3 2 1

  21. More Graphs Lie type B2 q9=-1 D(S3) Lie type B3 q12=-1 Extra colors for different objects…

  22. Classify Modular Categories Rankof an MC: # of simple objects Conjecture (Z. Wang 2003): The set { MCs of rank M } isfinite. Verified for: M=1, 2 [Ostrik], 3 and 4 [ER, Stong, Wang]

  23. Analogy Theorem (E. Landau 1903): The set { G : |Rep(G)|=N } is finite. Proof: Exercise (Hint: Use class equation)

  24. Classification by Graphs Theorem: (ER, Stong, Wang) Indecomposable, self-dual MCs of rank<5 are determined and classified by:

  25. Physical Feasibility Realizable TQC Bn action Unitary i.e. Unitary Modular Category

  26. Two Examples Unitary, for some q Never Unitary, for any q Lie type G2 q21=-1 “even part” for Lie type B2 q9=-1 For quantum group categories, can be complicated…

  27. General Problem G discrete, (G) U(N) unitary irrep. What is the closure of (G)? (modulo center) • SU(N) • Finite group • SO(N), E7, other compact groups… Key example: i(Bn)  U(Hom(Xn,Xi))

  28. Braid Group Reps. • Let X be any object in a unitary MC • Bn acts on Hilbert spaces End(Xn) as unitary operators: F(b), ba braid. • The gate set: {F(bi)}, bi braid generators.

  29. Computational Power {Ui} universal if {promotions ofUi} U(kn) qubits: k=2 Topological Quantum Computer universal Fi(Bn)denseinSU(Ni)

  30. Dense Image Paradigm Class #P-hard Link invariant F(Bn)dense Universal Top. Quantum Computer Eg. FQHE at =12/5?

  31. Property F A modular category D has property F if the subgroup: F(Bn)  GL(End(Vn)) is finite for all objects V in D.

  32. Example 1 Theorem: F(sl2, q ,L)has property Fif and only if L=2,3,4 or6. (Jones ‘86, Freedman-Larsen-Wang ‘02)

  33. Example 2 Theorem: [Etingof,ER,Witherspoon] Rep(DG) has property F for any finite group Gand 3-cocycle . More generally, true for braidedgroup-theoreticalfusion categories.

  34. Finite Group Paradigm Modular Cat. with prop.F Poly-time Link invariant Non-Universal Top. Quantum Computer Abelian anyons, FQHE at =5/2? quantum error correction?

  35. Categorical Dimensions For modular category D define dim(X) = TrD(IdX) =  R dim(D)=i(dim(Xi))2 IdX dim[End(Xn)]  [dim(X)]n

  36. Examples • In Rep(DG) all dim(V) • In F(sl2, q ,L), dim(Xi) = For L=4or6, dim(Xi) [L/2], for L=2or3, dim(Xi)  sin((i+1)/L) sin(/L)

  37. Property F Conjecture Conjecture: (ER) Let D be a modular category. Then D has property Fdim(C). Equivalent to: dim(Xi)2  for all simple Xi.

  38. Observations • Wang’s Conjecture is true for modular categories with dim(D) (Etingof,Nikshych,Ostrik) • My Conjecture would imply Wang’s for modular categories with property F.

  39. Current Problems • Construct more modular categories (explicitly!) • Prove Wang’s Conj. for more cases • Explore Density Paradigm • Explore Finite Image Paradigm • Prove Property FConjecture

  40. Thanks!

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