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Properties of Modular Categories and their Computation Consequences

Explore the realm of modular categories and their computational implications through an in-depth study of topological quantum field theories, topological states, quantum computation, and more. Discover the significance of modular categories in linking, knot theory, Hopf algebras, and symmetry analysis in topological states of matter.

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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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