Classical ( r , δ ) -locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum ( r , δ ) -locally recoverable codes which are quantum error-correcting codes capable of correcting δ − 1 qudit erasures from sets of at most r + δ − 1 qudits. We give a necessary and sufficient condition for a quantum stabilizer code Q ( C ) to be ( r , δ ) -locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code C used for constructing Q ( C ) and its symplectic dual C ⊥ s . When Q ( C ) comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of ( r , δ ) -local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.
Classical ( r , δ ) -locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum ( r , δ ) -locally recoverable codes which are quantum error-correcting codes capable of correcting δ − 1 qudit erasures from sets of at most r + δ − 1 qudits. We give a necessary and sufficient condition for a quantum stabilizer code Q ( C ) to be ( r , δ ) -locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code C used for constructing Q ( C ) and its symplectic dual C ⊥ s . When Q ( C ) comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of ( r , δ ) -local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.
Quantum (r,δ)-locally recoverable codes are quantum error-correcting codes capable of correcting δ−1 qudit erasures within one subset of qudits of cardinality at most r+δ−1. In this paper, we introduce the more general framework of entanglement-assisted quantum (r,δ)-locally recoverable codes, assuming that the local recovery operation is assisted by receiver-held qudits that remain unaffected by erasures.
We establish necessary and sufficient conditions for these codes to satisfy this property. For codes derived from Hermitian or Euclidean constructions, we establish connections between entanglement-assisted quantum and classical notions of (r,δ)-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum (r,δ)-locally recoverable codes from several families of classical codes, including bivariate J-affine variety codes, BCH codes, and homothetic-BCH codes.
Optimal pure quantum (r,δ)-locally recoverable codes from matrix-product construction
Locally recoverable codes (LRCs) are classical error-correcting codes widely used in large-scale distributed and cloud storage systems. Quantum locally recoverable codes of locality (r,δ) (quantum (r,δ)-LRCs) are the quantum counterpart of classical (r,δ)-LRCs. They allow us to correct erasures at several positions using a trace-preserving quantum operation acting on qudits of a larger set of positions. Quantum (r,δ)-LRCs, Q(C), can be constructed from classical Euclidean (or Hermitian) dual-containing codes C, and their recovery abilities are upper bounded by the minimum distance of the Euclidean (or Hermitian) dual of those codes. Parameters and localities of pure quantum (r,δ)-LRCs satisfy a Singleton-like bound; codes attaining equality are referred to as optimal. We consider matrix-product codes (MPCs) C and give constituent (or defining) matrices and conditions on the constituent codes such that the codes C satisfy the conditions to provide quantum (r,δ)-LRCs. As a consequence, we are able to determine their locality and parameters. Furthermore, we determine families of optimal pure quantum (r,δ)-LRCs derived from them.
Quantum (r,δ)-Locally Recoverable BCH and Homothetic-BCH Codes
Quantum (r,δ)-locally recoverable codes ((r,δ)-LRCs) are the quantum version of classical (r,δ)-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum (r,δ)-LRC, Q(C), can be constructed from an (r,δ)-LRC, C, which is Euclidean or Hermitian dual-containing.
This article is devoted to studying how to get quantum (r,δ)-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum (r,δ)-LRCs which are optimal for the Singleton-like bound.
Bounds for General (r,δ)-Quantum Locally Recoverable Codes
We study general (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=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−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=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>δ, which simultaneously attains all these bounds.
A classical linear code C of length n is said to have symbol locality (r,δ) if for any index j there exists a repair group Jj⊆{1,…,n} with j∈Jj and ∣Jj∣≤r+δ−1 such that any δ−1 or fewer erasures in Jj can be corrected by using codeword symbols only in Jj. Later it turned out that this way of defining r overestimates the number of necessary codeword symbols for multiple-erasure correction, and information locality was proposed to define r as the dimension of the punctured code of C onto Jj. Recently locality (r,δ) was proposed for quantum error-correcting codes by following the original definition of symbol locality (r,δ). We propose a quantum counterpart of the information locality for quantum stabilizer codes constructed by Hermitian orthogonality, and a linear algebraic procedure computing a smaller repair group predicted by the proposed information locality and simultaneously reducing the number of measured observables in decoding to its minimum possible value. Then we demonstrate that the previously proposed definition of quantum locality (r,δ) has the same drawback of overestimating the number of necessary codeword symbols for erasure correction by providing an explicit example of a quantum stabilizer code. Finally, we will give another example of a quantum stabilizer code constructed by Euclidean orthogonality and two different linear codes, with which a natural translation of the classical information locality into the quantum setting underestimates the number of necessary codeword symbols for erasure correction.
Constructions of Quantum (r,δ)-LRCs from cyclic codes
Classical (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)-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)-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.