Braiding defects in topological stabiliser codes of any dimension cannot be universal. (arXiv:1811.11789v2 [quant-ph] UPDATED)

Braiding defects in topological stabiliser codes has been widely studied as a
promising approach to fault-tolerant quantum computing. We present two no-go
theorems that place very strong limitations on the potential of such schemes
for universal fault-tolerant quantum computing in any spatial dimension. In
particular, we show that all logical operators implemented by braiding defects
in topological stabiliser codes are in the Clifford group, regardless of
dimension, and therefore cannot be universal. Moreover, supplementing braiding
of defects with locality-preserving logical operators (a generalisation of
transversal gates to topological codes) still cannot achieve a universal gate
set in any topological stabiliser code.

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