Locality-Preserving Logical Operators in Topological Stabiliser Codes

TL;DR

This paper presents a comprehensive method to identify all locality-preserving logical operators in a broad class of topological stabiliser codes, enhancing fault-tolerance analysis in quantum error correction.

Contribution

It introduces a systematic procedure linking locality-preserving operators to gapped domain walls, applicable to various topological codes including surface, colour, and quantum double models.

Findings

The procedure fully characterizes logical operators for stacked surface codes.

Application examples include codes with different boundary conditions and excitations.

The method generalizes to higher-dimensional and fermionic topological codes.

Abstract

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.