Two-block and multi-block hyperbolic algebra codes

We present a new family of quantum LDPC codes that extend group-algebra-based constructions to the setting of hyperbolic geometry. Building on the two-block group algebra (2BGA) codes of Lin and Pryadko (arXiv:2306.16400), we generalize the underlying algebraic framework by replacing finite group algebras with those derived from the translation groups of hyperbolic lattices. This yields families of hyperbolic quantum codes with improved distance-to-rate trade-offs compared to hyperbolic codes with local stabilizers. Furthermore, we construct hyperbolic multi-block codes inspired by Abelian multi-cycle codes (Lin et al., arXiv:2506.16910), extending their single-shot error-correction properties to non-abelian translation groups. Despite the non-abelian structure, we show that the redundant low-weight stabilizers enabling single-shot decoding persist for some of the hyperbolic translation groups. Our results demonstrate that hyperbolic lattices naturally support group-algebra-type LDPC code families with scalable parameters, bridging algebraic and geometric quantum code design.