Physics – Quantum Physics
Scientific paper
2010-12-24
Physics
Quantum Physics
minor edits, results unchanged
Scientific paper
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 $\textit{F}(H)$ 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 charge, flux, and 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 suggested for universal quantum computation: $S_3$ and $A_5$, and discover some interesting results and symmetries in the tables. The tables demonstrate that the anyons in physical theories based on $S_3$ and $A_5$ are all Majorana, but this is not the case for all finite groups. In addition, $SO(3)_4$ (the restriction of Chern-Simons theory $SU(2)_4$) and its mirror image are discovered as 3-particle subsystems in the $S_3$ quantum double. Finally, the probabilities of obtaining specific fusion products in quantum computation applications are determined and programmed in MAGMA. In the appendices, a few other non-abelian groups that may be of interest - $S_4$, $A_4$, and $D_4$ - are included. Throughout, connections to possible experiments are mentioned.
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