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"Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code", Yuri Lensky, Kostyantyn Kechedzhi, Igor Aleiner, Eun-Ah Kim, 2022

Reviewed December 29, 2023

Citation: Lensky, Yuri D., et al. "Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code." Annals of Physics 452 (2023): 169286.

Web: https://arxiv.org/abs/2210.09282

Tags: Error-correcting-codes, Ising-anyons, Toric-code, Defects/boundaries

This paper lays out a concrete approach towards implementation of Ising anyons on a quantum processor. The idea is to begin with the toric code, presented in Kitaev quantum double way on a generic graph whose vertices are all degree 2/3/4. One then adds orientation on all of the edges, with the condition that every plaquette has an odd number of counter clockwise edges. This is a sort of frustration condition on orientations known as the Kasteleyn condition, and serves as a key element of this theory. Flipping the orientations of all the edges touching a vertex maintains the Kasteleyn condition. Hence, the Kasteleyn condition induces a local Z2 action on vertices. That is, we get a Z2 Kasteleyn gauge field.

This Kasteleyn gauge field can be made physical by adding Majorana degrees of freedom at vertices. In particular, after adding the Majorana degrees of freedom one obtains Wilson loops which can measure the Kasteleyn flux in a region. It is shown that degree 3 vertices host Kasteleyn flux defects. These defects are shown to obey Ising fusion rules, and can be coherently manipulated to make braids. Thus, this results in a nice lattice implementation of Ising anyons.

Philosophically, it seems that the Kasteleyn gauge field corresponds to charge-flux symmetry, and the fact that its twist defects behave like Ising anyons is a coronary of the general fact that gauging the charge-flux symmetry in the toric code results in the Ising model.