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

TL;DR

This paper constructs quantum LDPC codes with fixed degrees that asymptotically reach the Gilbert-Varshamov bound, demonstrating their effectiveness in quantum error correction.

Contribution

It introduces a new construction of quantum LDPC codes from specific classical codes that achieve the GV bound at fixed degrees.

Findings

Codes maintain positive rate as blocklength grows.

Both code distances stay bounded away from zero with high probability.

Explicit degree choices attain the classical GV bound.

Abstract

We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. In the fixed-degree regime, 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 determine exactly which even regular degree choices in our construction attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.