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Algebraic Aspects of Topological Quantum Computing Eric Rowell Indiana University Collaborators Z. Wang (IU and MS Research*) M. Larsen (IU) R. Stong (Rice) * “Project Q” with M. Freedman, A. Kitaev, K. Walker and C. Nayak What is a Quantum Computer? Any system for computation based on quantum mechanical phenomena Create Manipulate Measure Quantum Systems Classical vs. Quantum • Bits {0,1} 1 • Qubits: V=C2 1 (superposition) 0 • Logical Operations on {0,1}n • Unitary Operations on V X n • Deterministic: output unique • Probabilistic: output varies (Uncertainty principle) a 0 Anyons: 2D Electron Gas 1011 electrons/cm2 particle exchange 9 mK fusion defects=quasi-particles 10 Tesla Topological Computation Computation Physics output measure apply operators braid initialize create particles Algebraic Characterization Anyonic System Top. Quantum Computer Modular Categories Toy Model: Rep(G) • Irreps: {V1=C, V2,…,Vk} • Sum V + W, product V X W, duals W* • Semisimple: every W= + miVi • Rep: Sn EndG(V X n) Braid Group Bn “Quantum Sn” Generated by: 1 i i+1 n i=1,…,n-1 bi = Multiplication is by concatenation: = Concept: Modular Category deform group G Rep(G) Sn action Modular Category Bn action Axiomatic definition due to Turaev Modular Category • Simple objects {X0=C,X1,…XM-1} + Rep(G) properties • Rep. Bn End(X X n) (braid group action) • Non-degeneracy: S-matrix invertible Dictionary: MCs vs. TQCs Simple objects Xi Bn-action Elementary particle types Operations (unitary) X0 =C Vacuum state X0 Xi X Xi* Creation Constructions of MCs quantum group g Uqg Lie algebra Rep(Uqg) semisimplify F |q|=1 Survey: (E.R. Contemp. Math.) (to appear) Also, quantum double G D(G) Rep(D(G)) finite group 13 Physical Feasibility Realizable TQC Bn action Unitary Uqg Unitarity results: (Wenzl 98), (Xu 98) & (E.R. 05) Computational Power Physically realizable {Ui} universal if all {Ui} = { all unitaries } TQC universal F(Bn) dense in PkSU(k) Results: in (Freedman, Larsen, Wang 02) and (Larsen, E.R., Wang 05) Physical Hurdle: Realizable as Anyonic Systems? Classify MCs 1-1 Recall: distinct particle types Simple objects in MC Conjecture (Z. Wang 03): The set { MCs of rank M } is finite. Classified for: M=1, 2 (V. Ostrik), 3 and 4 (E.R., Stong, Wang) True for finite groups! (Landau 1903) Groethendieck Semiring • Assume X=X*. For a MC D: Xi X Xj = + Nijk Xk • Semiring Gr(D):=(Ob(D), + , X ) • Encoded in matrices (Ni)jk := Nijk Modular Group • Non-dengeneracy S symmetric • Compatibility T diagonal • 1 1 0 1 T, 0 -1 1 0 S give a unitary projective rep. of SL(2,Z) Our Approach • Study Gr(D) and reps. of SL(2,Z) • Ocneanu Rigidity: Finite-to-one MCs {Ni} • Verlinde Formula: {Ni} determined by S-matrix Some Number Theory • Let pi(x) = det(Ni - xI) and K = Split({pi},Q). • Study Gal(K/Q): always abelian! • Nijk integers, Sij algebraic, constraints polynomials. Sketch of Proof (M<5) 1. Show: 1 Gal(K/Q) SM 2. Use Gal(K/Q) + constraints to determine (S, {Ni}) 3. For each S find T rep. of SL(2,Z) 4. Find realizations. Graphs of MCs • Simple Xi multigraph Gi : Vertices labeled by 0,…,M-1 Nijk edges j k • Question: What graphs possible? Example (Lie type G2, q10=-1) Rank 4 MC with fusion rules: N111=N113=N123=N222=N233=N333=1; N112=N122= N223=0 G1: G2: G 3: 0 1 0 2 2 0 3 1 3 3 2 1 Tensor Decomposable! Classification by Graphs Theorem: (E.R., Stong, Wang) Indecomposable, self-dual MCs of rank<5 are classified by: Future Directions • Classification of all MCs • Prove Wang’s conjecture • Images of Bn reps… • Connections to: link/manifold invariants, Hopf algebras, operator algebras… Thanks!