Beyond the toric code: engineering quantum gates with non-Abelian topological order

Topological phases of matter provide a natural setting for protecting quantum information by encoding it in a degenerate ground-state subspace. However, protecting quantum information is only part of the challenge: quantum computation also requires manipulating the encoded information without sacrificing its topological protection. In the toric code, a paradigmatic example of topological order, the quantum gates that can be implemented with topological protection are severely restricted.

 

In this talk, I will discuss how non-Abelian topological order can be used to overcome these restrictions. I will first introduce group surface codes, which generalize the toric code using non-Abelian finite groups and suitable boundary conditions. These codes enable protected quantum gates that cannot be implemented natively in the toric code. I will then describe irrep surface codes, which use the representation theory of finite groups to systematically construct topological codes supporting prescribed sets of quantum gates. Finally, I will discuss how these codes can be interfaced with conventional codes, providing a route to quantum computation that exploits the richer structure of topological phases of matter.