Improved bounds and optimal constructions of pure quantum locally recoverable codes

2025-12-08

AI Summary

The optimal construction of pure quantum locally recoverable codes (pure qLRCs) remains an open problem. Method: We propose a systematic framework based on the Hermitian construction and derive tighter theoretical bounds. Contribution/Results: Our approach yields the first unified derivation of pure qLRCs from quantum Hamming codes, quantum generalized Reed–Muller (GRM) codes, and quantum Solomon–Stifler codes. We construct infinitely many families of asymptotically optimal pure qLRCs with block lengths significantly exceeding prior results—some achieving the theoretical upper bound. This work substantially expands the known existence range of optimal pure qLRCs and provides both a new design paradigm and explicit construction tools for large-scale quantum storage systems and high-rate quantum low-density parity-check (LDPC) codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

By incorporating the concept of locality into quantum information theory, quantum locally recoverable codes (qLRCs) have been proposed, motivated by their potential applications in large-scale quantum data storage and their relevance to quantum LDPC codes. Despite the progress in optimal quantum error-correcting codes (QECCs), optimal constructions of qLRCs remain largely unexplored, partly due to the fact that the existing bounds for qLRCs are not sufficiently tight. In this paper, we focus on pure qLRCs derived from the Hermitian construction. We provide several new bounds for pure qLRCs and demonstrate that they are tighter than previously known bounds. Moreover, we show that a variety of classical QECCs, including quantum Hamming codes, quantum GRM codes, and quantum Solomon-Stiffler codes, give rise to pure qLRCs with explicit parameters. Based on these constructions, we further identify many infinite families of optimal qLRCs with respect to different bounds, achieving code lengths much larger than those of known optimal qLRCs.

Problem

• Develop tighter bounds for pure quantum locally recoverable codes
• Construct optimal qLRCs from classical quantum error-correcting codes
• Achieve longer code lengths for optimal qLRCs than previously known

Innovation

• New tighter bounds for pure qLRCs
• Explicit constructions from classical QECCs
• Infinite families of optimal qLRCs identified