Venkatesan Guruswami

Erasure recovery is a fundamental aspect of classical data storage, especially in distributed systems where data redundancy and local repair are essential for reliability. Extending these ideas to the quantum domain provides valuable insights for developing robust and scalable quantum memories. In quantum information theory, erasures—where the positions of lost qubits are known—are generally easier to correct than arbitrary errors, which require both detection and correction. This distinction simplifies recovery protocols and enhances fault tolerance. Recent experimental advances have demonstrated that detected quantum errors can be converted into erasures in physical systems such as neutral-atom–based and superconducting qubit architectures, highlighting the practical importance of studying quantum erasure recovery. 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 qubit to be reconstructed using only a small subset of other qubits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering $(r,\delta)$-QLRCs characterized by locality parameter $r$ and local distance $\delta \geq 2$. We present constructions of both random and explicit $(r,\delta)$-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.