Information, Geometry, and Physics Seminar

We will share some recent results that are instructive for approaching the classification of 3D TQFTs and topological order by computational complexity.

We explain how the Turaev-Viro-Barrett-Westbury state-sum invariants that arise from Tambara-Yamagami categories are efficient to compute for 3-manifolds, provided there is a bound on their first Betti number. On the one hand, this is a pretty good algorithm given the fact that these TQFT invariants are #P-hard to compute for the smallest member of the family of Tambara-Yamagami categories (and should be hard to compute more generally). On the other hand, it isn't too surprising given that Tambara-Yamagami categories are only a slight generalization of the finite dimensional representation category of a finite abelian group, whose associated TVBW invariants are easy to compute. In any case, one can interpret our parametrized algorithm to mean that our inability to classically compute these quantum invariants in polynomial time is due to the fact that 3-manifolds can have a large first Betti number.

This talk is based on joint work with Clément Maria and Eric Samperton.