Universal quantum computation with group surface codes

The surface code is one of the leading approaches to building a fault-tolerant quantum computer. A central challenge for the surface code is implementing non-Clifford operations -- unitaries which map Pauli operators to non-Pauli operators. We introduce group surface codes, which are a natural generalization of the usual Z₂ surface code and can be understood as quantum double models of finite groups with specific boundary conditions. We show that group surface codes, for suitably chosen groups, can be leveraged to perform non-Clifford gates in the Z₂ surface code. We describe three strategies for completing a universal gate set using group surface codes: through magic state preparation, using transversal non-Clifford gates, and by sliding group surface codes. These strategies extend recent efforts in performing universal quantum computation in topological orders without the braiding of anyons. Moreover, they show how certain group surface codes allow us to bypass the restrictions set by the Bravyi-König theorem, which only limits the computational power of topological (Pauli) stabiliser models.