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arXiv:2512.07256 · IEEE Transactions on Information Theory

Improved bounds and optimal constructions of pure quantum locally recoverable codes

Yang Li, Shitao Li, Gaojun Luo, San Ling

Published Jan 1, 2026• 0 citations
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Abstract

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.

DOI
10.1109/tit.2026.3734348

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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 et al.et~al.et al. (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}t\in \{2,3\}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.
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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.
Bounds for General (r,δ)(r,δ)(r,δ)-Quantum Locally Recoverable Codes
We study general (r,δ)(r,δ)(r,δ)-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When δ=2δ=2δ=2, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to δ−2δ-2δ−2. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and d=δd=δd=δ, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and d>δd>δd>δ, which simultaneously attains all these bounds.
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.