Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabiliser codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including colour codes and toric codes, as well as more general codes such as abelian quantum double models and codes with fermionic excitations in more than two dimensions.
We analyse a model for fault-tolerant quantum computation with low overhead
suitable for situations where the noise is biased. The basis for this scheme is
a gadget for the fault-tolerant preparation of magic states that enable
universal fault-tolerant quantum computation using only Clifford gates that
preserve the noise bias. We analyse the distillation of $|T\rangle$-type magic
states using this gadget at the physical level, followed by concatenation with
the 15-qubit quantum Reed-Muller code, and comparing our results with standard
constructions. In the regime where the noise bias (rate of Pauli $Z$ errors
relative to other single-qubit errors) is greater than a factor of 10, our
scheme has lower overhead across a broad range of relevant noise rates.Comment: 9 pages, 6 figures, comments welcome; v2 published versio
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