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"Anyon computers with smaller groups", Carlos Mochon, 2004

Reviewed July 26, 2023

Citation: Mochon, Carlos. "Anyon computers with smaller groups." Physical Review A 69.3 (2004): 032306.

Web: https://arxiv.org/abs/quant-ph/0306063

Tags: Foundational, Kitaev-quantum-double

This paper shows that the quantum double model based on any non-nilpotent group gives universal quantum computation by braiding and fusion. This improves on the earlier work

> Mochon, Carlos. "Anyons from nonsolvable finite groups are sufficient for universal quantum computation." Physical Review A 67.2 (2003): 022315.

The earlier paper had only shown universal quantum computation for non-solvable groups. As is always true, as your group becomes more abelian you need to use more recourses. In the non-solvable case you only had to braid one class of anyon together (i.e. those corresponding to conjugacy classes) and you could leave the other class (i.e. those corresponding to irreducible representations) alone. In this paper, they fundamentally use the relationship/interplay between the two classes of anyons. This gives much more power, but necessarily also comes with much more complexity.

The smallest non-nilponent group is S3, of order 6. This happens to also be the smallest non-abelian group, and hence there is no hope of performing universal quantum computation by braiding on a smaller group.

This is not to say that S3 is the simplest non-abelian group to perform TQC with. The larger dihedral group D4 was the first non-abelian phase to be implemented in the lab:

> Iqbal, Mohsin, et al. "Creation of Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor." arXiv preprint arXiv:2305.03766 (2023).

Apparently, like all things in these early days, the algorithm for S3 was proposed by Kitaev. The algorithm presented here is especially simple for the case of S3, and Mochon treats it with extra care. However, this paper is still quite hard to read and many people are skeptical of it. Hence, a better reference to this result is Zhenghan's paper:

> Cui, Shawn X., Seung-Moon Hong, and Zhenghan Wang. "Universal quantum computation with weakly integral anyons." Quantum Information Processing 14 (2015): 2687-2727.