We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual $\mathbb{C}(\mathbb{Z}_p)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a $n$ dimensional vector space which we call $H_n$. The $\mathbb{Z}_p$ gauge particles act on the vertex particles and thus $H_n$ can be thought of as a $\mathbb{C}(\mathbb{Z}_p)$ module. An exactly solvable model is built with operators acting in the corresponding Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of $n$ and $p$, though we believe this feature holds for all $n>p$. We will see that non-Abelian anyons of the quantum double of $\mathbb{C}(S_3)$ are obtained as part of the vertex excitations of the model with $n=6$ and $p=3$. Ising anyons are obtained in the model with $n=4$ and $p=2$. The $n=3$ and $p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than $\mathbb{Z}_p$. This makes them possible candidates for realizing quantum computation.
Universal Gates from Braiding and Fusing Anyons on Quantum Hardware
Harvard University（哈佛大学）；Quantinuum；Stony Brook University（石溪大学）；C. N. Yang Institute for Theoretical Physics, Stony Brook University（石溪大学杨振宁理论物理研究所）；University of Chicago（芝加哥大学）
Unveiling Topological Fusion in Quantum Hall Systems from Microscopic Principles
Jagiellonian University（雅盖隆大学）；Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与工程中心）；Princeton University（普林斯顿大学）；Karlstads Universitet（卡尔斯特大学）；Institute for Advanced Study, Princeton（普林斯顿高等研究院）；Indiana University（印第安纳大学）
