Citation: Freedman, Michael H., and Zhenghan Wang. "Large quantum Fourier transforms are never exactly realized by braiding conformal blocks." Physical Review A—Atomic, Molecular, and Optical Physics 75.3 (2007): 032322.
Web: https://arxiv.org/abs/cond-mat/0609411
Tags: TQFT, Fibonacci-anyons, No-go
In this lovely paper, the authors show that there is no exact way of implementing the quantum fourier transform for large N using a quantum field computer. The idea is that running Shor's factoring algorithm requires these large fourier transforms, so it would be nice to be able to run these fourier transforms exactly (like you can in the quantum circuit model) and not approximately. The hope, as the authors put it, is that maybe "the arithmetic properties of Fibonacci anyons could be matched up to the number theory of factoring" to make an exact algorithm. The answer of this paper is a no-go theorem: no such exact implementation exists.
The point is that the coefficients in the quantum fourier transform are roots of unity of large order. The authors demonstrate that the mapping class group representations can all be normalized so that their imagine lives in some fixed number field K, and it is a general result that the number of roots of unity in a given number field is always finite (proof example: Dirichlet's unit theorem). As such, there is no exact algorithm for the Fourier transform at all but finitely many N.
This paper is notable as one of the first fruitful applications of number theory (or more generally number-theoretic techniques) to topological quantum information.
This paper also highlights a tension in the normalization of MTCs in regards to number theory. It is always possible to normalize an MTC so that all of its structure data lives in an abelian number field, however it is NOT always possible to normalize an MTC so that all of its structure data lives in an abelian number field AND its F-matrices are unitary.