Qudit Twisted-Torus Codes in the Bivariate Bicycle Framework

TL;DR

This work extends the Laurent-polynomial, translation-invariant CSS framework for twisted-torus codes to qudits over finite fields, motivated by evidence that twisting improves finite-size performance in the BB code family. By encoding qudit stabilizers with two sparse Laurent polynomials $f$ and $g$ and imposing twisted boundary identifications $(\,alpha, \,beta, \,gamma)$, the authors compute the number of logical qudits via Gröbner-basis methods and estimate distances with randomized schemes, focusing on a weight-6 ansatz. The key finding is that twisted-torus qudit codes typically achieve larger distances than untwisted counterparts and surpass prior twisted-qubit instances in the same regime, with several high $k d^{2}/n$ codes tabulated. This suggests that higher local dimension combined with twisted identifications can improve finite-length rate–distance tradeoffs in translation-invariant LDPC families, offering a viable path toward more robust, scalable quantum codes. Future work includes extending the ansatz, tightening distance certification, and analyzing decoding performance under realistic noise models.

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.