Quantum Hierarchical Locally Recoverable Codes

2026-06-04

AI Summary

This work addresses the challenge of qubit loss by introducing Quantum Hierarchical Locally Recoverable Codes (QHLRCs), which for the first time enable a multi-level local recovery mechanism supporting an arbitrary number of hierarchy layers \( h \geq 2 \). Leveraging the CSS framework, dual-containing classical codes, and a quantum adaptation of the Tamo–Barg construction, the authors establish a Singleton-type bound for QHLRCs and provide both explicit and random code constructions. An efficient decoding algorithm is also devised, substantially reducing storage overhead and recovery latency. This study offers a novel theoretical framework for quantum erasure correction and advances the development of fault-tolerant quantum storage architectures.

Abstract

Quantum locally recoverable codes (QLRCs) have recently gained attention as a framework for achieving efficient quantum storage with local recovery capabilities. Analogous to their classical counterparts, QLRCs allow a lost qudit to be reconstructed using only a small subset of other qudits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering $(r,δ)$ QLRCs characterized by locality parameter $r$ and local distance $δ\geq 2$. We present constructions of both random and explicit $(r,δ)$ QLRCs, including explicit families based on the quantum Tamo--Barg construction. We also present an efficient decoding algorithm for these quantum Tamo--Barg codes. Furthermore, we introduce quantum \emph{hierarchical} locally recoverable codes (QHLRCs), which extend local recovery to multiple hierarchical levels. For any integer $h\geq 2$, we construct both random and explicit $h$-level QHLRCs, the latter being $h$-level quantum Tamo--Barg codes, and establish a Singleton-like bound for these codes using a CSS framework built from dual-containing classical codes. These results advance the theoretical foundations of quantum erasure recovery and contribute to the design of efficient quantum storage architectures.

Problem

• quantum locally recoverable codes
• hierarchical recovery
• quantum storage
• local recovery
• quantum erasure recovery

Innovation

• quantum hierarchical locally recoverable codes
• quantum Tamo–Barg construction
• local recovery
• CSS codes
• Singleton-like bound