Certified exact distances for new binary quantum codes with quasi-cyclic structure
Ancestor Lifting of Quasi-Cyclic Additive Codes and Certified Quantum Codes
Information Theory
Summary
Quantum computers need special codes to protect their fragile data from errors. The authors studied specific families of codes called quasi-cyclic additive codes and developed ways to measure their exact error-correcting ability. They used new mathematical methods to certify the quality of these codes, breaking records in known quantum code tables. Their work also supplies explicit constructions for codes that previously had no known form. This advances reliable design of quantum error-correcting codes for future quantum computing.
What this means in practice
- •For quantum hardware engineers: Integrate certified high-performance quantum error-correcting codes into quantum processors to improve error tolerance and operational reliability.
- •For quantum communication designers: Use exact-distance quantum stabilizer codes for reliable information transmission over noisy quantum channels, supporting advanced quantum networks.
Authors
Liangdong Lu, Ran Xu, Ruipan Yang, Qiang Fu
Abstract
Additive codes over $\Ffour$ that are self-orthogonal under the trace inner product correspond to binary quantum stabilizer codes. We study 1-generator $(1,2)$-quasi-cyclic (QC) additive codes and their exact quantum distances. For composite lengths $n=\ell m$, an ancestor-lifting theorem transports dual words of the folded ancestor to the length-$n$ code. A two-layer decomposition for $\ell=2$ and a CRT splitting for odd $\ell$ yield dimension laws and explain equality, escape, and mixed-branch behavior in the certified examples. A palindromic criterion ensures symplectic self-orthogonality, while reversal pairing and a symplectic Gram rank formula describe broader construction families. Exact distances are established by information-set decoding screening, Brouwer--Zimmermann computation on the additive dual, and independent witness verification. We certify twenty-one record-breaking binary quantum codes across twenty entries of Grassl's code table, with twenty-eight certified members in total. Propagation improves thirty-four further entries, and explicit constructions are supplied for three table-tying GV entries marked Code Missing Construction. Six additional candidates are reported only with rigorous upper bounds and heuristic screening evidence. The results connect divisor-lattice structure in additive quasi-cyclic codes with reproducible certification of exact quantum distances.