Braid Group Representations from Braiding Gapped Boundaries of Dijkgraaf-Witten Theories

TL;DR

This paper investigates braid group representations from gapped boundaries in Dijkgraaf-Witten theories, showing they are finite and monomial, which limits their use in universal topological quantum computing.

Contribution

It provides explicit formulas for monomial matrices and ground state degeneracies, revealing the non-universality of braiding gapped boundaries for quantum gates.

Findings

All braid representations are finite groups.

Explicit formulas for monomial matrices are derived.

Gapped boundaries alone cannot achieve universal quantum gates.

Abstract

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 resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. 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.