Braid group representations from braiding gapped boundaries of Dijkgraaf-Witten theories (2017)
In Topological Quantum Computation, quantum gates are implemented by representations of the braid group, B_n, on spaces of morphisms in a modular category C. For a given group G and a 3-cocycle \omega, images of that representation on C = Z(Vec^\omega_G) are finite, but is not known in general what specific gates can be obtained. A family of algebras in Z(G, \omega) called Lagrangian Algebras are of particular physical interest. They are denoted L[H, \gamma], where H is a subgroup and \gamma is a 2-cocycle on H. We show that the spaces Hom_{Z(G,\omega)}(1, L[H, \gamma]^n) have a canonical structure of monomial spaces and that with respect to this structure, the representation of B_n is monomial. We calculate the nonzero entries of these matrices and use this information to show how they can be used to implement a CNOT gate.