Applications of quantum representations of the mapping class group to quantum computers Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in  and 1 and 2.
The following sources 3 and 4  imply that there are applications of quantum representations of the mapping class group to quantum computing. 
What are some of these applications? Which concrete mathematical questions (e.g. from low dimensional topology or quantum algebra) have applications/are relevant to quantum computing?
 A: The context is topological quantum computation, where quantum information is stored nonlocally in a physical system, so that it is protected from decoherence by local sources of noise. The nonlocal degree of freedom is a socalled non-Abelian anyon, a particle-like excitation which is described by a (2+1)-dimensional topological quantum field theory. Arxiv:1705.06206 provides a survey of the mathematics of topological quantum computing, with a list of conjectures and open problems.
I should add, as a physicist, that I am uncertain whether any of these mathematical questions are relevant in the quest to actually build and operate a quantum computer. The key challenge there is to identify a physical system that has these exotic particles. The fractional quantum Hall effect was a primary candidate for several decades, but this system has now been largely abandoned in favor of superconducting systems, where the energy gaps can be much larger (allowing for operation in a realistic temperature range). Microsoft is heavily invested in the design of a topological quantum computer using anyons in the Ising universality class (Majorana fermions). These are "trivial" from a mathematical perspective, since they only implement a Clifford algebra and do not provide access to the full unitary group. There are no realistic options for Fibonacci anyons (which would cover all unitary operations).
