I will give an overview of some relations between finite group theory, G-equivariant topological quantum field theory, and the computational complexity of invariants of 3-manifolds, both classical and quantum. We will start with one of the simplest kinds of invariants in knot theory: the coloring invariants, introduced by Fox when giving a talk to undergraduates in the 1950s. We will then build up to the idea of G-equivariant TQFT (aka homotopy QFT with target K(G,1)), which mathematically describes the topological order determined by a symmetry-enriched topological phase of matter. Physicists have studied these in part motivated by the search for new universal topological quantum computing architectures.
Our goal will be to convey two complexity-theoretic lessons. First, when G is sufficiently complicated (nonabelian simple), the simple-to-define coloring invariants associated to G are, in fact, very difficult to compute, even on a quantum computer. Second, no matter what finite group G one uses, a 3-dimensional G-equivariant TQFT can not be used for universal topological quantum computation if the underlying non-equivariant theory is not already universal. This talk is based on joint works with Greg Kuperberg and Colleen Delaney.
https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09