Classical locally recoverable codes, which permit highly efficient recovery from localized errors as well as global recovery from larger errors, provide some of the most useful codes for distributed data storage in practice. In this paper, we initiate the study of quantum locally recoverable codes (qLRCs). In the long term, like their classical counterparts, such qLRCs may be used for large-scale quantum data storage. Furthermore, our results have concrete implications for quantum LDPC codes, which are widely applicable to near-term quantum error-correction, as local recoverability is a weakening of the LDPC property.
Classical locally recoverable codes, which permit highly efficient recovery from localized errors as well as global recovery from larger errors, provide some of the most useful codes for distributed data storage in practice. In this paper, we initiate the study of quantum locally recoverable codes (qLRCs). In the long term, like their classical counterparts, such qLRCs may be used for large-scale quantum data storage. Furthermore, our results have concrete implications for quantum LDPC codes, which are widely applicable to near-term quantum error-correction, as local recoverability is a weakening of the LDPC property.
Classical locally recoverable codes, which permit highly efficient recovery from localized errors as well as global recovery from larger errors, provide some of the most useful codes for distributed data storage in practice. In this paper, we initiate the study of quantum locally recoverable codes (qLRCs). In the long term, like their classical counterparts, such qLRCs may be used for large-scale quantum data storage. Our results also have concrete implications for quantum LDPC codes, which are applicable to near-term quantum error-correction. After defining quantum local recoverability, we provide an explicit construction of qLRCs based on the classical LRCs of Tamo and Barg (2014), which we show have (1) a close-to-optimal rate-distance tradeoff (i.e. near the Singleton bound), (2) an efficient decoder, and (3) permit good spatial locality in a physical implementation. Although the analysis is significantly more involved than in the classical case, we obtain close-to-optimal parameters by introducing a "folded" version of our quantum Tamo-Barg (qTB) codes, which we then analyze using a combination of algebraic techniques. We furthermore present and analyze two additional constructions using more basic techniques, namely random qLRCs, and qLRCs from AEL distance amplification. Each of these constructions has some advantages, but neither achieves all 3 properties of our folded qTB codes described above. We complement these constructions with Singleton-like bounds that show our qLRC constructions achieve close-to-optimal parameters. We also apply these results to obtain Singleton-like bounds for qLDPC codes, which to the best of our knowledge are novel. We then show that even the weakest form of a stronger locality property called local correctability, which permits more robust local recovery and is achieved by certain classical codes, is impossible quantumly.
By incorporating the concept of locality into quantum information theory, quantum locally recoverable codes (qLRCs) have been proposed, motivated by their potential applications in large-scale quantum data storage and their relevance to quantum LDPC codes. Despite the progress in optimal quantum error-correcting codes (QECCs), optimal constructions of qLRCs remain largely unexplored, partly due to the fact that the existing bounds for qLRCs are not sufficiently tight. In this paper, we focus on pure qLRCs derived from the Hermitian construction. We provide several new bounds for pure qLRCs and demonstrate that they are tighter than previously known bounds. Moreover, we show that a variety of classical QECCs, including quantum Hamming codes, quantum GRM codes, and quantum Solomon-Stiffler codes, give rise to pure qLRCs with explicit parameters. Based on these constructions, we further identify many infinite families of optimal qLRCs with respect to different bounds, achieving code lengths much larger than those of known optimal qLRCs.
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 r governing recovery from both bit-flip (X-type) and phase-flip (Z-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}, in which X-type and Z-type errors are corrected from recovery sets of size at most rX+1 and rZ+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 and C2 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.
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 etal. (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 t∈{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.
We introduce the concept of quantum locally recoverable codes (qLRCs) with intersecting recovery sets. We derive a singleton-like bound for these codes by leveraging the additional information provided by the intersecting recovery sets. Furthermore, we provide a construction for qLRCs with intersecting recovery sets by introducing a variation of the hypergraph product. Finally, we apply our qLRC methods to obtain improved results for classical LRCs. These results may provide new insights into the locality of quantum error correction code.
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≥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≥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.