Towards Topological Quantum Computation? - Knotting and Fusing Flux Tubes

摘要: Models for topological quantum computation are based on braiding and fusing anyons (quasiparticles of fractional statistics) in (2+1)-D. The that can exist a physical theory determined by the symmetry group Hamiltonian. In case Hamiltonian undergoes spontaneous breaking full G to finite residual gauge H, particles given representations double $D(H)$ subgroup. quasi-triangular Hopf Algebra D(H) is obtained from Drinfeld's construction applied algebra F(H) functions H. A major new contribution this work program written MAGMA compute (and their properties - including spin) system with an arbitrary group, addition fusion rules those particles. We explicitly two non-abelian doubles suggested universal computation: $S_3$ $A_5$, discover some interesting results, subsystems, symmetries tables. SO(3)_4 (the restriction Chern-Simons $SU(2)_4$) its mirror image discovered as 3-particle subsystems 8-particle double. tables demonstrate both $A_5$ all Majorana, but not groups. appendices, remaining nonabelian subgroups SO(3) $S_4$, $A_4$, $D_4$ second infinite family $D_n$) tabulated analyzed. addition, probabilities obtaining any product applications programmed MAGMA. Throughout, connections possible experiments mentioned.