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This research presents a significant theoretical advance by constructing finite-degree quantum LDPC codes that asymptotically achieve the Gilbert-Varshamov bound, offering improved error-correcting capabilities without increasing circuit complexity. The construction relies on nested CSS code pairs derived from specific combinatorial designs, providing rigorous proofs for bounded coding rates and relative distances as blocklength grows. These results are critical for advancing the practical feasibility of fault-tolerant quantum computing architectures that require high-density, efficient error correction.
Finite-Degree Quantum LDPC Codes Reaching the Gilbert-Varshamov Bound
arXiv:2603.24588v3 Announce Type: replace Abstract: We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. For fixed balanced triples with even component degrees and $j_Z\ge4$, we prove that the coding rate stays bounded away from zero and that the relative distances on both sides stay bounded away from zero with probability tending to one as the blocklength grows. Moreover, within an explicit low-degree search window, we prove that all 56 balanced triples in that window with even common row degree, column degree at least four, and positive design quantum rate attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.
Abstract:We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. For fixed balanced triples with even component degrees and $j_Z\ge4$, we prove that the coding rate stays bounded away from zero and that the relative distances on both sides stay bounded away from zero with probability tending to one as the blocklength grows. Moreover, within an explicit low-degree search window, we prove that all 56 balanced triples in that window with even common row degree, column degree at least four, and positive design quantum rate attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.
55 pages, 5 figures. Expanded proofs and exact-rational certificates; corrected theorem scope and enumerator estimates; added actual-rate analysis; revised the certified-degree figure. Data and code: this https URL
Quantum Physics (quant-ph); Information Theory (cs.IT)
arXiv:2603.24588 [quant-ph]
(or arXiv:2603.24588v3 [quant-ph] for this version)
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