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arXiv:2606.06736

Quantum Hierarchical Locally Recoverable Codes

Venkatesan Guruswami, Rutuja Kshirsagar, Pranav Trivedi

Published Jun 4, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2606.06736

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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)(r,δ)(r,δ) QLRCs characterized by locality parameter rrr and local distance δ≥2δ\geq 2δ≥2. We present constructions of both random and explicit (r,δ)(r,δ)(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≥2h\geq 2h≥2, we construct both random and explicit hhh-level QHLRCs, the latter being hhh-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.
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In this work, we study (r,t,x)(r,t,x)(r,t,x) quantum locally recoverable codes (qLRCs) with locality rrr, ttt recovery sets per qudit, and intersection parameter xxx. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an (r,t,x)(r,t,x)(r,t,x)-qLRC if and only if the underlying classical codes are (r,t,x)(r,t,x)(r,t,x) classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing (r,t,x)(r,t,x)(r,t,x)-cLRCs, which yield binary (r,t,x)(r,t,x)(r,t,x)-qLRCs via the CSS construction. For CSS (r,t,x)(r,t,x)(r,t,x)-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.
Constructions of Quantum (r,δ)(r,δ)(r,δ)-LRCs from cyclic codes
Classical (r,δ)(r,δ)(r,δ) locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic (r,δ)(r,δ)(r,δ)-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of (r,δ)(r,δ)(r,δ)-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.
Locally recoverable codes (LRCs) permit reconstruction of a lost coordinate from a small subset of the remaining coordinates, a property that is a fundamental coding technique for distributed storage and, more recently, for fault-tolerant quantum memories and distributed storage systems. Existing constructions of quantum locally recoverable codes (qLRCs) and their entanglement-assisted generalizations (EA-qLRCs) impose a single locality parameter rrr governing recovery from both bit-flip (XXX-type) and phase-flip (ZZZ-type) errors, even though physical noise is rarely symmetric between the two. We introduce entanglement-assisted quantum locally recoverable codes (EA-qLRC) with \emph{asymmetric locality} {rX,rZ}\{r_X,r_Z\}{rX​,rZ​}, in which XXX-type and ZZZ-type errors are corrected from recovery sets of size at most rX+1r_X+1rX​+1 and rZ+1r_Z+1rZ​+1, respectively. We extend a stabilizer sufficient condition for this property and, in particular, the CSS framework, where asymmetric locality corresponds to separate locality conditions on the two classical codes C1\mathcal C_1C1​ and C2\mathcal C_2C2​ underlying the construction. From this characterization we derive Singleton-, Griesmer-, Plotkin-, sphere-packing-, and Cadambe\textendash Mazumdar-like bounds for CSS-type asymmetric EA-qLRCs (AEA-qLRCs) built from Proposition~\ref{CSS-EA-qLRC-Asym}, and we characterize exactly when a pure CSS-type AEA-qLRC built from a pair of classical LRCs attains the Singleton-like bound with equality.
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Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality rrr and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to ℓ\ellℓ-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.