2010/12/24 by Meagan B. Thompson, Thompson, Meagan B. · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Anyon #FOS: Mathematics #FOS: Physical sciences #Finite group #Gauge group #Gauge theory #Group (periodic table) #Hamiltonian (control theory) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Mathematics #Other Condensed Matter (cond-mat.other) #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Quantum computer #Quantum mechanics #Representation Theory (math.RT) #Theoretical physics #Topological quantum computer #cond-mat.other #hep-th #math-ph #math.MP #math.RT #quant-ph
paper · pdf · doi:10.48550/arxiv.1012.5432
published in arXiv (Cornell University) (Cornell University) · minor edits, results unchanged
openalex publication_date 2010/12/24 · arxiv created 2012/12/26 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Models for topological quantum computation are based on braiding and fusing\nanyons (quasiparticles of fractional statistics) in (2+1)-D. The anyons that\ncan exist in a physical theory are determined by the symmetry group of the\nHamiltonian. In the case that the Hamiltonian undergoes spontaneous symmetry\nbreaking of the full symmetry group G to a finite residual gauge group H,\nparticles are given by representations of the quantum double D(H) of the\nsubgroup. The quasi-triangular Hopf Algebra D(H) is obtained from Drinfeld's\nquantum double construction applied to the algebra F(H) of functions on the\nfinite group H.\n A major new contribution of this work is a program written in MAGMA to\ncompute the particles (and their properties - including spin) that can exist in\na system with an arbitrary finite residual gauge group, in addition to the\nbraiding and fusion rules for those particles. We compute explicitly the fusion\nrules for two non-abelian group doubles suggested for universal quantum\ncomputation: S3 and A5, and discover some interesting results,\nsubsystems, and symmetries in the tables. SO(3)4 (the restriction of\nChern-Simons theory SU(2)4) and its mirror image are discovered as\n3-particle subsystems in the 8-particle S3 quantum double. The tables\ndemonstrate that both S3 and A5 anyons are all Majorana, but this is not\nthe case for all finite groups. In the appendices, the quantum doubles for the\nremaining nonabelian subgroups of SO(3) - S4, A4, and D4 (the second\nin the infinite family Dn) - are tabulated and analyzed. In addition, the\nprobabilities of obtaining any given fusion product in quantum computation\napplications are determined and programmed in MAGMA. Throughout, connections to\npossible experiments are mentioned.\n