Topological Codes from Space Groups: A Route beyond Translation Invariance

Topological codes, including the toric code, are among the most important classes of stabilizer codes. Existing constructions and analyses of such codes, however, overwhelmingly assume translation invariance. Here we introduce a framework for constructing Calderbank–Shor–Steane (CSS) codes based on space groups which combine translations with point-group operations, thereby breaking translation invariance. To characterize these codes, we develop a module-theoretic approach based on invariant theory that provides a rigorous criterion for topological order and enables the computation of the number of independent anyon types. Although the inclusion of point-group operations might naively appear to hinder practical implementation, we find that these codes can actually exhibit enhanced locality compared to their purely translation-invariant counterparts. Our framework thus broadens the landscape of topological codes and opens new avenues for their co-design with quantum computing platforms.