Topological Codes from Space Groups: A Route beyond Translation Invariance

Abstract

Translation invariance underlies all algebraic constructions of topological codes with geometrical locality. It has remained an open question whether codes that generically break this invariance can still be topological and simultaneously possess geometrical locality. Resolving this question is important both fundamentally---deepening our understanding of topological phases---and practically, as relaxing translation invariance could vastly expand the design space and potentially reduce resource overhead in fault-tolerant architectures. Here we introduce space-group codes, in which crystallographic point-group operations enter the bulk stabilizer algebra; bivariate bicycle (BB) codes arise as the translation-only limit. The key insight is that the point-group orbit resolves topology and locality together: it yields a computable algebraic criterion for topological order and a folded geometry in which point-group operations become local. We identify space-group codes whose code parameters exceed the reported same-blocklength, same-check-weight BB benchmarks. In five parameter-matched neutral-atom comparisons, reflection codes reduce the optimized movement cost in every case, by up to $60\%$, while folded placements also enable lower-overhead multilayer superconducting layouts. Treating spatial operations as a code-design variable therefore opens a route to topological codes jointly optimized for information protection and hardware geometry.

24 pages, 6 figures, update the abstract and introduction and correct some typos