Category theory and topological quantum computing

Transcription

1 Categor theor and topological quantum computing Gregor Schaumann Group seminar QOS Freiburg 7..3

2 Introduction Conformal field theor Invariants of manifolds and knots Topological field theor Tensor categories and higher categories Tft A robust frameork for quantum theories of geometr Topological Quantum Computing Fault-tolerant codes in -dimensional sstems possible (A. Kitaev) Quantum gravit Quantum information + Quantum Gravit Gravit in + dimensions is topological, tfts describe a quantum theor (E. Witten) Categories Crucial ingredient

3 Table of contents Topological quantum computing Quantum computing Anons Constructing topological models The code spaces and tfts Boundaries and defects

4 General requirements for quantum computing Basic quantum information: Qudit, i.e. a vector in the Hilbertspace C d, d. Encoding of k qudits: Unitar embedding (C d ) k H, H phsical Hilbert space. Computation via universal gates: Phsical transformations {A i } on H that approimate ever unitar in U(d k ). In practice one needs at least one entangling gate A: there is a product state Φ Φ... Φ k ith A( Φ Φ... Φ k ) not a product. easurements on the sstem can project to single qudits.

5 Problems for a quantum computer Decoherence: Interaction ith the environment leads to mied states. Error correction: Encoding and processing leads to errors. In addition there are thermal fluctuations. Idea: Use a robust hardare that is insensitive to an local perturbation. Cannot encode qudits locall! Need phsical sstems ith a global/topological interaction.

6 Statistics in QFTs Construct a quantum field theor: Assign to an initial N-particle configurations a Hilbertspace H I. To a connected piece in the trajector space assign an operator H I H F. In 3d, an echange of identical particles leads to a phase: Ψ(, ) e iθ Ψ(, ). Double echange is homotop to the identit onl bosons (θ = ) and fermions (θ = π) possible. In d, the double echange is not topological trivial braid statistics (i.e. an θ) are possible.

7 The braid group The braid group B N ith N strands is generated b σ i, i =,..., N and relations: N σ, N σ i σ i+ σ i = σ 3 N i+ σ i σ i Anons transform under representations of the braid groups. B Z and σ e iθ is a d representation.

8 Anon tpes Abelian anons m different particle tpes, θ ab a, b =,..., m phases for braiding particle a around b. Non-abelian anons A configuration of N identical particles has a degeneration space of dimension g >. In a Basis Φ α, α =,..., g, the echange σ is represented b a matri ρ(σ ) αβ. Braiding particle and : Φ α ρ(σ ) αβ Φ β, braiding particle and 3: Φ α ρ(σ ) αβ Φ β, non-abelian: [ρ(σ ), ρ(σ )]. Onl a model ith non-abelian anons can have an entangling gate using braiding!

9 Fusion of anons Representations of B N depend on particle tpe and number. To anons together are again an anon. Fied particle number m requires fusion channels Φ a Φ b = c N ab c Φ c, N ab c N. There is the vacuum e and ever particle Φ a has an antiparticle Φ a such that Φ a Φ a fuse to the vacuum. Associative and commutative fusion ring. Non-abelian anons: At least one fusion process must have different channels.

10 Eample The fusion ring of SU() has three particle tpes e, Φ and fusion channels Φ Φ = e + Φ, Φ Φ = Φ, Φ Φ = e. and Φ To dimensional fusion space for fusing three Φ -particles to one Φ -particle, Basis:,, or,, () Change of basis: l ( ) k = l k

11 Consistenc requirements additional data: F-matri and R-matri. Conceptual description desirable to obtain a Basis -independent version of the data, identif equivalent F-, R-matrices, formulate theories independent of Basis, describe submodels, relations beteen different models.

12 odular categor A modular categor C is a coherent description of possible braid statistics ith a finite number of particle tpes. The data for C include: a set of objects,,... C, for an to objects, a finite dimensional vector spaces Hom(, ), a tensor product, : C C C (a functor), for an three objects isomorphisms F(,, z) : ( ) z ( z) (natural isomorphisms), for to objects isomorphisms R(, z) : Intuition: Categorification Algebra elements product associative commutative odular categor objects F R

13 Kitaev s toric code, Z/Z-model Consider a k k-lattice on the torus. Assign k -qubits, one to each edge. Hilbertspace H. Consider operators: for each verte v, A v = σj : for each plaquette p, B p = j around v j around p Hamilton operator H = v A v f B f, groundspace {Ψ H, HΨ = Ψ} has dimension four. σ z j : v p Ecitations String operators S z (s) = σj z j s and S (t) = j t σ j generate electric and magnetic anons e and m (abelian). t z s z

14 Generalization Z/Z G: Group model G finite group. Assign qudits in C G to edges of the lattice. For a verte v and a group element g define operators A g v, A g v v z = g g v gz, u gu For a plaquette p define the operator B p B p p z = δ e,uz p z, v u Define ith A v = G g G Ag s and the Hamiltonian H G = v A v p B p. Groundspace H Ψ = Ψ : Discretized version of flat G-connections. This reproduces the toric code for G = Z/Z. v u

15 Further generalizations G C: string model, the state spaces Without ecitations: Consider a closed surface Σ and a fied fusion categor C. Associate a Hilbertspace H Σ of graphs: embed directed 3-valent graphs in Σ, label edges b alloed simple objects of C, f.e. k n i l m Identif graphs related b the folloing local relations: j i m j l k i = i i = diδ je = F ij m kln i l n j k

16 Turaev-Viro, the 3d amplitudes ΣF Let be a 3-manifold connecting the -manifolds Σ I and Σ F ΣI The Turaev-Viro Tft associated to an operator Z () : H ΣI H ΣF, as follos: Take to string states Ψ I and Ψ F in H I and H F, living on triangulations of Σ I and Σ F. Choose a triangulation of etending the triangulations of its boundar. For an labeling l of the edges of a tetrahedron t, assign the number F(t, l) = Fijk mno, multipl and average to obtain Ψ F Z () Ψ I = F(t, l). l t

17 Boundaries Different fusion categories C and D correspond to different phases of a bigger theor if there is a defect separating the phases. C D odel for defect alls Suppose the defect has ecitations described b a mathematical object, the fusion of a bulk ecitation to the defect suggests that there is an action C, is a module categor over C, a representation.

18 boundar ecitations Defect alls can have defect lines: C D µ Defect lines are functors µ : N. N Bulk ecitations are defect lines of the trivial defect all! Eistence of Braiding. H ρ μ ν K N

19 A general frameork for defects Since defect lines can have pointlike defects e arrive at a three laer structure, here each laer has coherent duals. D C µ ν ρ τ µ ρ ν τ Ψ Φ Φ Ψ H N H N µ ν Φ N η ν Ω N µ ν Φ N η Ψ a) b) c) µ ν Φ N K D B C D C C C D D C C C C D D D µ ν Φ N C D K B

20 Conclusion Open questions Ho can e incorporate defects in topological quantum computation? [Shor et al.]: condensations at boundaries temperature dependent effects at defects? Hamiltonian descriptions of defects? Full defect model formulation, etend [Kitaev, Kong].

21 [Wilczek: Fractional Statistics and Anon Superconductivit] Introdution to anons, from the path integral perspective. [Kitaev: Fault-tolerant quantum computation b anons] Definition of the toric code and the G-model [Levin, Wen: String-net condensation: A phsical mechanism for topological phases] Definition of the string model [Turev, Viro: State sum invariants of 3-manifolds and quantum 6jsmbols] Definition of the 3d tft [Kitaev, Kong: odels for gapped boundaries and domain alls] Defects in string model [Barrett, eusburger, S: Gra categories ith duals and their diagrams] Abstract language for defects in 3d tfts