Qudit Twisted-Torus Codes in the Bivariate Bicycle Framework

We study finite-length qudit quantum low-density parity-check (LDPC) codes from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Recent qubit work [PRX Quantum 6, 020357 (2025)] showed that, within the bivariate-bicycle viewpoint, twisting generalized toric patterns can significantly improve finite-size performance as measured by $k d^{2}/n$. Here $n$ denotes the number of physical qudits, $k$ the number of logical qudits, and $d$ the code distance. Building on this insight, we extend the search to qudit codes over finite fields. Using algebraic methods, we compute the number of logical qudits and identify compact codes with favorable rate–distance tradeoffs. Overall, for the finite sizes explored, twisted-torus qudit constructions typically achieve larger distances than their untwisted counterparts and outperform previously reported twisted qubit instances. The best new codes are tabulated.

💡 Research Summary

This paper investigates finite‑length qudit quantum low‑density parity‑check (LDPC) codes constructed from translation‑invariant CSS designs placed on two‑dimensional tori with twisted boundary conditions. Building on recent qubit work that demonstrated substantial finite‑size improvements when generalized toric patterns are twisted, the authors extend the concept to qudits of prime dimension q, working over the finite field 𝔽_q.

The theoretical framework uses the Laurent‑polynomial ring R_q = 𝔽_q