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"Classification of topologically protected gates for local stabilizer codes", Sergey Bravyi, Robert Konig, 2013

Reviewed August 18, 2023

Citation: Bravyi, Sergey, and Robert König. "Classification of topologically protected gates for local stabilizer codes." Physical review letters 110.17 (2013): 170503.

Web: https://arxiv.org/abs/1206.1609E

Tags: Abelian-anyons, No-go

This paper gives a general restriction on what gates can be fault-tolerantly implemented in "topological" codes. That is, codes which can be implemented with O(1) connections on a regular lattice. The power of the model depends only on the dimension of the lattice. If the lattice is two dimensional then all of the gates will be Clifford, and if the lattice is on a higher-dimensional system then the gates will be on a higher level of the Clifford hierarchy.

This is useful because it gives a conceptual picture of why the Clifford group is so important: that's just how topological codes work. Additionally, this gives motivation for the fact that when you do things based on a finite group G then there should be a "universal" model of what gates are powerful, namely, the qu-G-it Clifford group. That is, one might expect a generalization of this Bravyi-König theorem.