Qudit Twisted-Torus Codes in the Bivariate Bicycle Framework

TL;DR

This paper explores twisted-torus qudit LDPC codes on two-dimensional tori, demonstrating improved finite-size performance and larger distances compared to untwisted and previous qubit codes, through algebraic construction methods.

Contribution

It extends the twisted-torus code framework to qudits over finite fields, providing algebraic methods to identify codes with better rate-distance tradeoffs.

Findings

Twisted-torus qudit codes outperform untwisted counterparts in code distance.

New codes achieve larger distances than previous twisted qubit codes.

Algebraic methods enable systematic identification of compact, high-performance codes.

Abstract

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.