On optimal quantum LRCs from the Hermitian construction and $t$-designs

2025-08-19

AI Summary

This paper addresses the open problem of parameter bounds and constructions for quantum locally recoverable codes (qLRCs) (Luo et al., 2025), proposing a novel framework based on the Hermitian construction. The core method integrates *t*-design theory into the construction of near-MDS (NMDS) classical codes, yielding dual-containing classical LRCs that are then transformed into qLRCs via the Hermitian duality. Key contributions include: (1) establishing four new upper bounds on qLRC parameters; (2) constructing the first infinite families of NMDS codes supporting *t*-designs; and (3) obtaining three infinite families of optimal qLRCs with more flexible parameters—each achieving tightness simultaneously under all four derived bounds. This approach overcomes fundamental limitations of conventional CSS-based constructions and significantly expands the parameter space of known optimal qLRCs.

Technology Category

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

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

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $tin {2,3}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

Problem

• Construct optimal quantum locally recoverable codes
• Solve open problem on Hermitian construction bounds
• Provide flexible parameters for quantum data storage

Innovation

• Hermitian construction for optimal qLRCs
• NMDS codes supporting t-designs
• Flexible dual-containing classical LRCs