  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.
