Citation: Freedman, Michael, et al. "Topological quantum computation." Bulletin of the American Mathematical Society 40.1 (2003): 31-38.
Web: https://arxiv.org/abs/quant-ph/0101025
Tags: Foundational, Expository, Pedagogical, TQFT
This is an expository paper, which brings together all of the foundational work into a coherent narrative. Freedman's original quest for extreme computational power has been proven to be impossible, but Kitaev's quest for fault-tolerance is alive and well, and it is this fault-tolerance which is put in the forefront of the narrative.
This paper is a great way of getting to know what TQC was like in 2003 and getting into Freedman-Kitaev's head, but of course it is by now very outdated. There are lots of nice philosophical tidbits thrown in throughout, especially in the introduction. For instance, Freedman proposed a new thesis along the lines of Turing-Church's. "All reasonable computational models which add the resources of quantum mechanics (or quantum field theory) to classical computation yield (efficiently) inter-simulable classes: there is one quantum theory of computation".
Another fun quote, in reference to the fact that certain TQFTs can approximate the Jones polynomial at roots of unity: "This seeming curiosity is actually the tip of an iceberg which links quantum computation both to low dimensional topology and the theory of anyons; the motion of anyons in a two dimensional system defines a braid in 2 + 1 dimension. This iceberg is a model of quantum computation based on topological, rather than local, degrees of freedom".