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Citation: Bravyi, Sergey, et al. "Adaptive constant-depth circuits for manipulating non-abelian anyons." arXiv preprint arXiv:2205.01933 (2022).
Topological quantum computing presents itself as a method for fault tolerant quantum computation. For this to be realized, one needs to either create topological phases of matter, or come up with algorithms to implement topological order on a quantum processor. This paper makes massive strides in the question of algorithms for quantum processors.
The previous state of the art came 13 years prior, in the article
> Brennen, G. K., Miguel Aguado, and J. Ignacio Cirac. "Simulations of quantum double models." New Journal of Physics 11.5 (2009): 053009.
these algorithms were expensive, and their depth scales linearly with the size of the lattice. The major improvement of the article at hand is showing that, when G is solvable, the computations can be done in constant-depth. This is especially relevant given the fact that solvable non-nilpotent groups like S3 are sufficient for universal quantum computation by topological methods.
This article is long, and goes into fantastic detail. So fantastic that it was used as the basis of the first quantum processor simulation of non-abelian topological order:
> Iqbal, Mohsin, et al. "Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor." arXiv preprint arXiv:2305.03766 (2023).