Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes

Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realised by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher dimensional defect-braiding schemes have the potential to play an important role in realising fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher dimensional codes, specifically: in self-dual surface codes; the three dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.