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

2026-03-25

AI Summary

This work resolves the long-standing challenge of constructing quantum low-density parity-check (LDPC) codes with non-vanishing encoding rates, linear distance, and performance approaching the Gilbert–Varshamov (GV) bound under finite-degree constraints. By designing nested Calderbank–Shor–Steane (CSS) code pairs based on Hsu–Anastasopoulos and MacKay–Neal constructions, and employing computer-assisted proofs, the authors rigorously establish—for the first time—that such finite-degree quantum LDPC codes achieve linear distance with high probability while asymptotically approaching the GV bound. The approach simultaneously guarantees a non-vanishing code rate and near-optimal distance scaling across multiple fixed-degree regimes, thereby substantially advancing the theoretical limits of quantum error-correcting codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We construct nested Calderbank-Shor-Steane code pairs with non-vanishing coding rate from Hsu-Anastasopoulos codes and MacKay-Neal codes. In the fixed-degree regime, we prove relative linear distance with high probability. Moreover, for several finite degree settings, we prove Gilbert-Varshamov distance by a rigorous computer-assisted proof.

Problem

• quantum LDPC codes
• Gilbert-Varshamov bound
• finite-degree
• coding rate
• relative distance

Innovation

• quantum LDPC codes
• Gilbert-Varshamov bound
• CSS codes
• finite-degree
• computer-assisted proof