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Abstract
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We study representations of the braid groups from braiding gapped boundaries of
Dijkgraaf–Witten theories and their twisted generalizations, which are (twisted)
quantum doubled topological orders in two spatial dimensions. We show that the
braid representations associated to Lagrangian algebras are all monomial with
respect to some specific bases. We give explicit formulas for the monomial matrices
and the ground state degeneracy of the Kitaev models that are Hamiltonian
realizations of Dijkgraaf–Witten theories. Our results imply that braiding gapped
boundaries alone cannot provide universal gate sets for topological quantum
computing with gapped boundaries.
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Keywords
topological quantum computation, modular categories, braid
group
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Mathematical Subject Classification 2010
Primary: 18D10, 20C35, 20F36
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Milestones
Received: 12 July 2017
Revised: 21 August 2018
Accepted: 17 September 2018
Published: 20 July 2019
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