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Towards Topological Quantum Computation? - Knotting and Fusing Flux Tubes | | Meagan B. Thompson
; Frank A. Wilczek
; | | Date: |
24 Dec 2010 | | 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 $ extit{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, closed subsystems are analyzed with a view
towards topological quantum computation. The probabilities of obtaining
specific fusion products in quantum computation schemes are determined for
theories based on finite groups. The MAGMA program includes a procedure to
determine the probabilities for any finite group based on these results. 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. | | Source: | arXiv, 1012.5432 | | Services: | Forum | Review | PDF | Favorites | |
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