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.

翻译：量子局部可恢复码（QLRCs）近期作为实现高效量子存储并具备局部恢复能力的框架而受到关注。与传统经典编码类似，QLRCs允许仅通过少量其他量子位（qudit）的子集重构丢失的量子位，从而降低恢复过程中的资源与操作开销。本文通过考虑具有局部性参数 $r$ 和局部距离 $\delta \geq 2$ 的 $(r,\delta)$ QLRCs，扩展了对QLRCs的研究。我们提出了随机显式 $(r,\delta)$ QLRCs的构造方案，包括基于量子Tamo-Barg构造的显式编码族，并给出了这些量子Tamo-Barg码的高效解码算法。此外，我们引入了量子层级局部可恢复码（QHLRCs），该编码将局部恢复扩展至多个层级。对任意整数 $h \geq 2$，我们构造了随机与显式 $h$ 层QHLRCs（后者为 $h$ 层量子Tamo-Barg码），并通过基于对偶包含经典码的CSS框架建立了这类码的Singleton型界限。这些结果推进了量子擦除恢复的理论基础，并为高效量子存储架构的设计提供了支持。