Towards Topological Quantum Computation? - Knotting and Fusing Flux Tubes

Abstract

Models for topological quantum computation are based on braiding and fusing anyons (quasiparticles of fractional statistics) in (2+1)-D. The anyons that can exist in a physical theory are determined by the symmetry group of the Hamiltonian. In the case that the Hamiltonian undergoes spontaneous symmetry breaking of the full symmetry group G to a finite residual gauge group H, particles 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. 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 group doubles suggested for universal quantum computation: $S_3$ and $A_5$, and discover some interesting results, subsystems, and symmetries in the tables. SO(3)_4 (the restriction of Chern-Simons theory $SU(2)_4$) and its mirror image are discovered as 3-particle subsystems in the 8-particle $S_3$ quantum double. The tables demonstrate that both $S_3$ and $A_5$ anyons are all Majorana, but this is not the case for all finite groups. In the appendices, the quantum doubles for the remaining nonabelian subgroups of SO(3) - $S_4$, $A_4$, and $D_4$ (the second in the infinite family $D_n$) - are tabulated and analyzed. In addition, the probabilities of obtaining any given fusion product in quantum computation applications are determined and programmed in MAGMA. Throughout, connections to possible experiments are mentioned.