Citation: Samperton, Eric. "Topological quantum computation is hyperbolic." Communications in Mathematical Physics 402.1 (2023): 79-96.
Web: https://arxiv.org/abs/2201.00857
Tags: Computer-scientific
In this paper, the author prove the following theorem. Suppose that C is some input modular category, and that K is an oriented ribbon knot. There is an efficient algorithm which outputs an oriented ribbon knot diagram K' such that K' is "nice" and K and K' have the same Witten-Reshetikhin-Turaev invariants based on the modular category C. The philosophical takeaway is that the niceness conditions on K' don't help you solve the problem.
Specifically the niceness conditions are as follows: K' is an alternating knot (meaning that traveling along it you will alternate from overcrossing and underdressing), K' has minimal crossing number over all diagrams of equivalent knots, and K' is a hyperbolic knot (meaning that its knot complement has a complete hyperbolic structure).
The original hope of the author seems to be that hyperbolic knots might admit some extra computational resource which would make their invariants easier to calculate. This was wrong. The problem to computing invariants is not made any easier by restricting to the sorts of nice knots.