Generalized toric codes on twisted tori for quantum error correction

Understand

The Kitaev toric code is widely considered one of the leading candidates for error correction in fault-tolerant quantum computation.

• However, direct methods to increase its logical dimensions, such as lattice surgery or introducing punctures, often incur prohibitive overheads.
• In this work, we introduce a ring-theoretic approach for efficiently analyzing topological CSS codes in two dimensions, enabling the exploration of generalized toric codes with larger logical dimensions on twisted tori.
• Using Gr\"obner bases, we simplify stabilizer syndromes to efficiently identify anyon excitations and their geometric periodicities, even under twisted periodic boundary conditions.