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Towards Topological Quantum Computation? — Knotting and Fusing Flux Tubes

Meagan Thompson, **MIT**.
__Abstract:__
Models for topological quantum computation are based on braiding and fusing anyons (quasiparticles of fractional statistics) in 2D. The anyons that can exist in a physical theory are determined by the symmetry group of the Hamiltonian. From the mathematical perspective, any theory of anyons must have braiding and fusion rules that satisfy certain consistency conditions known as the Seiberg-Moore Polynomial Equations (also known as the pentagon and hexagon equations). Maclane's coherence theorem states these are in fact all that is required in order to achieve commutativity of all combinations of fusion and braiding (i.e. a consistent physical theory). Two applications of the Hexagon Equation yield the Yang-Baxter Equation familiar from statistical mechanics: σ_{j} σ_{j+1} σ_{j} = σ_{j+1} σ_{j} σ_{j+1} where the σ_{i} are the abstract braid group generators. It is an unsolved mathematical problem to determine in general all the matrix solutions to the Yang-Baxter Equation. In the case that the Hamiltonian undergoes spontaneous symmetry breaking of the full symmetry group G to a finite residual gauge group H, however, solutions are given by representations of the quantum double D(H) of the subgroup. The quasi-triangular Hopf Algebra D(H) is obtained from Drinfeld's quantum double construction applied to the algebra F(H) of functions on the finite group H. As a vector space, D(H) = F(H) ⊗ ℂ[H] = C(H × H) where ℂ[H] is the group algebra over the complex numbers and C(H × H) is the space of ℂ-valued functions on H × H.

A major new contribution of this work is a program written in MAGMA to compute the particles (and their properties—including spin) that can exist in a system with an arbitrary finite residual gauge group in addition to the braiding and fusion rules for those particles. We compute explicitly the fusion rules for two non-abelian groups thought to be sufficient for universal quantum computation under certain circumstances: S_{3} and A_{5}, and determine that the anyons are all Majorana for these groups. (In the appendices, a few other non-abelian groups of interest—S_{4}, A_{4}, and D_{4}—are addressed). In addition, experimental proposals for topological quantum computation with these groups are suggested, assessed, and compared to other quantum computing proposals currently on the table.