Finite-Degree Quantum LDPC Codes Reaching the Gilbert-Varshamov Bound

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 $j_Z\ge 4$, 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.

arXiv comment55 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: https://github.com/kasaikenta/finite-degree-gv-certificates