Twisted Quantum Codes Boost Error Correction and Extend Computing Potential

Researchers investigate finite-length qudit quantum low-density parity-check codes constructed using translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Mourad Halla, and colleagues demonstrate that twisting generalized toric patterns, viewed through a bivariate-bicycle framework, substantially improves finite-size performance. This work extends the search to qudit codes over finite fields, employing algebraic methods to compute qudit numbers and pinpoint compact codes exhibiting favourable rate-distance trade-offs. The findings reveal that, across the finite sizes examined, twisted-torus qudit constructions generally attain greater distances than untwisted codes and surpass previously published twisted instances, with the most promising new codes meticulously tabulated.

Finite-length qudit LDPC codes on twisted tori enhance quantum error correction performance significantly

Scientists are pioneering advancements in quantum error correction through the development of qudit codes on twisted tori, achieving improved performance over existing qubit instances. Recent work demonstrated that twisting generalized toric patterns significantly enhances finite-size performance, a concept now extended to qudits over finite fields.
This research focuses on finite-length qudit quantum low-density parity-check (LDPC) codes constructed from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. By employing algebraic methods, researchers compute the number of logical qudits and identify compact codes exhibiting favorable rate, distance tradeoffs.

The study builds upon the bivariate-bicycle viewpoint, revealing that twisting generalized toric patterns can substantially improve finite-size performance, measured by the ratio kd²/n, where n represents the number of physical qudits, k the number of logical qudits, and d the code distance. Extending this insight, the work explores qudit codes over finite fields, utilizing algebraic techniques to determine the number of logical qudits and pinpoint compact codes with advantageous rate-distance characteristics.
For the finite sizes investigated, twisted-torus qudit constructions consistently achieve larger distances compared to their untwisted counterparts, surpassing previously reported twisted qubit instances. Researchers leverage a Laurent-polynomial formalism to describe qudit CSS codes on twisted tori, working over finite fields and utilizing Gröbner-basis computations in SageMath to efficiently calculate the number of logical qudits.

Distances are estimated using the probabilistic algorithm within the GAP package QDistRnd, enabling a finite-length search within a weight-6 ansatz. The best performing code instances are meticulously tabulated, providing a valuable resource for the quantum computing community. This approach clarifies how finite-size behavior is strongly influenced by the choice of periodic boundary conditions, demonstrating that suitably twisted torus identifications can alter the logical dimension and yield superior finite codes.

These findings are particularly significant as quantum error-correcting codes are essential for constructing large-scale, fault-tolerant quantum computers. Topological stabilizer codes, like those explored in this work, encode quantum information into global degrees of freedom with local check operators, offering a promising route toward scalable fault tolerance.

The ability to systematically classify topological order and efficiently determine logical dimensions, facilitated by the algebraic viewpoint introduced by Haah, is a key innovation. This research not only advances the theoretical understanding of qudit codes but also provides concrete examples of high-performance codes with potential for implementation in future quantum technologies.

Construction and performance evaluation of twisted-torus qudit LDPC codes are presented

Finite-length qudit quantum low-density parity-check (LDPC) codes were investigated using translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. The research built upon prior work demonstrating that twisting generalized toric patterns enhances finite-size performance, specifically as measured by the ratio of code distance to the number of physical qudits.

Algebraic methods were central to this work, enabling the computation of the number of qudits required for these codes and the identification of compact codes exhibiting favorable rate, distance tradeoffs. The study employed twisted-torus qudit constructions to systematically explore configurations with and without twisting, assessing their performance across various finite sizes.

Codes were constructed on square lattices, and the impact of twisting boundary conditions was evaluated by comparing the resulting code distances. This approach allowed for the tabulation of the best new codes discovered, showcasing improvements over previously reported twisted instances. The work leveraged the Sage Mathematics Software System for algebraic calculations and code construction.

Distance certification was performed using the QDistRnd package within the GAP system, a tool designed for computing the minimum distance of quantum error-correcting codes. This ensured rigorous comparisons between different code families and parameter ranges. The research focused on identifying twists that not only optimize the (n, k, d) parameters, representing the number of qudits, the number of logical qudits, and the code distance, but also potentially improve decoding thresholds and runtime performance. These investigations contribute to a more systematic understanding of how twisted identifications interact with the algebraic data defining qudit codes, ultimately controlling logical dimension and distance.

Performance variation with differing code parameters and quantisation levels is expected and should be thoroughly tested

For codes with $q=3$ and $k=4$, the [[18,4,4]]q=3[[18, 4, 4]]_{q=3}[[18,4,4]]q=3​ code exhibits a performance metric of 5.56, while [[36,4,8]]q=3[[36, 4, 8]]_{q=3}[[36,4,8]]q=3​ achieves 7.11 and [[48,4,10]]q=3[[48, 4, 10]]_{q=3}[[48,4,10]]q=3​ reaches 8.33, demonstrating a clear increase in performance with larger code parameters. This trend continues for larger constructions, with the [[156,4,20]]q=3[[156, 4, 20]]_{q=3}[[156,4,20]]q=3​ code attaining a metric of 10.26, although a slight decrease is observed for [[222,4,23]]q=3[[222, 4, 23]]_{q=3}[[222,4,23]]q=3​, which achieves 9.53. For $k=6$, the [[28,6,7]]q=3[[28, 6, 7]]_{q=3}[[28,6,7]]q=3​ code yields a performance value of 10.50, followed by [[42,6,9]]q=3[[42, 6, 9]]_{q=3}[[42,6,9]]q=3​ at 11.57 and [[56,6,11]]q=3[[56, 6, 11]]_{q=3}[[56,6,11]]q=3​ reaching 12.96. The [[84,6,14]]q=3[[84, 6, 14]]_{q=3}[[84,6,14]]q=3​ code achieves 14.00, while [[112,6,16]]q=3[[112, 6, 16]]_{q=3}[[112,6,16]]q=3​ attains a slightly lower value of 13.71.

When $k=8$, performance improves further, with [[54,8,10]]q=3[[54, 8, 10]]_{q=3}[[54,8,10]]q=3​ providing 14.81, [[60,8,11]]q=3[[60, 8, 11]]_{q=3}[[60,8,11]]q=3​ yielding 16.13, and [[72,8,12]]q=3[[72, 8, 12]]_{q=3}[[72,8,12]]q=3​ reaching 16.00. Larger codes continue this upward trend: [[108,8,16]]q=3[[108, 8, 16]]_{q=3}[[108,8,16]]q=3​ achieves 18.96, [[162,8,20]]q=3[[162, 8, 20]]_{q=3}[[162,8,20]]q=3​ attains 19.75, and the largest example, [[198,8,24]]q=3[[198, 8, 24]]_{q=3}[[198,8,24]]q=3​, culminates in a performance metric of 23.27. Switching to $q=5$ with $k=4$, the [[24,4,7]]q=5[[24, 4, 7]]_{q=5}[[24,4,7]]q=5​ code yields 8.17, [[30,4,8]]q=5[[30, 4, 8]]_{q=5}[[30,4,8]]q=5​ gives 8.53, and [[36,4,8]]q=5[[36, 4, 8]]_{q=5}[[36,4,8]]q=5​ reaches 7.11. Performance improves for larger constructions, with [[48,4,11]]q=5[[48, 4, 11]]_{q=5}[[48,4,11]]q=5​ achieving 10.08, [[66,4,13]]q=5[[66, 4, 13]]_{q=5}[[66,4,13]]q=5​ attaining 10.24, and [[78,4,15]]q=5[[78, 4, 15]]_{q=5}[[78,4,15]]q=5​ also reaching a metric of 10.24.

Overall, these results demonstrate that employing twisted boundary identifications can significantly enhance finite-size behavior in quantum LDPC codes. The algebraic formulation over finite fields enables efficient computation of the number of logical qudits across a broad range of candidate patterns, particularly when combined with randomized decoding techniques.

Twisted boundary conditions optimise finite qudit LDPC code parameters for improved performance

Researchers have demonstrated improved finite-length qudit quantum low-density parity-check (LDPC) codes through the application of twisted boundary conditions on two-dimensional tori. These codes, constructed using translation-invariant CSS constructions, exhibit enhanced performance, as measured by the ratio of physical to logical qudits and code distance, compared to untwisted counterparts and previously reported instances.

The investigation involved algebraic computations to determine the number of logical qudits and identify compact codes with advantageous rate-distance characteristics for various finite sizes. Specifically, the study details several new codes with parameters [[n, k, d]]q, where n represents the number of physical qudits, k the number of logical qudits, d the code distance, and q the finite field size.

Codes were explored for q = 3 with k = 4, 6, and 8, as well as for q = 5 with k = 4 and 6, revealing instances with larger distances than previously known for similar parameters. The results indicate that twisting the toric patterns generally leads to better finite-size performance. The authors acknowledge that the computations were limited to finite sizes and specific parameter ranges.

Further research could explore the behaviour of these codes in the asymptotic limit and investigate the impact of different twisting configurations. Future work might also focus on extending these techniques to higher-dimensional codes or exploring alternative CSS constructions to optimise code performance and practicality for quantum error correction. These findings contribute to the development of more robust and efficient quantum codes, essential for building scalable and reliable quantum computers.