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} $\{r_X,r_Z\}$ , in which $X$ -type and $Z$ -type errors are corrected from recovery sets of size at most $r_X+1$ and $r_Z+1$ , 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 $\mathcal C_1$ and $\mathcal C_2$ 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.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} $\{r_X,r_Z\}$, in which $X$-type and $Z$-type errors are corrected from recovery sets of size at most $r_X+1$ and $r_Z+1$, 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 $\mathcal C_1$ and $\mathcal C_2$ 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.

quant-pharXiv:2610.23889Sep 29, 20260 citations

Two Families of Optimal Quantum Locally Recoverable Codes

Dengcheng Xie, Shixin Zhu, Zhonghua Sun

International Journal of Theoretical PhysicsMar 21, 20253 citations

Quantum $(r,δ)$ -Locally Recoverable BCH and Homothetic-BCH CodesQuantum $(r,δ)$-Locally Recoverable BCH and Homothetic-BCH Codes

Carlos Galindo, Fernando Hernando, Ryutaroh Matsumoto

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.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.

cs.ITCross-listedarXiv:2601.22567Feb 2, 20260 citations

Bounds and Constructions of Quantum Locally Recoverable Codes From Quantum CSS Codes

Gaojun Luo, Bocong Chen, Martianus Frederic Ezerman +1 more

Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential application in quantum data storage. In this paper, we use classical LRCs to investigate quantum LRCs. We prove that the parameters of quantum LRCs are bounded by their classical counterparts. We deduce bounds on the parameters of quantum LRCs from bounds on the parameters of the classical ones. We establish a characterization of optimal pure quantum LRCs based on classical codes with specific properties. Using well-crafted classical LRCs as ingredients in the construction of quantum CSS codes, we offer the first construction of several families of optimal pure quantum LRCs.Classical locally recoverable codes (LRCs) have become indispensable in distributed storage systems. They provide efficient recovery in terms of localized errors. Quantum LRCs have very recently been introduced for their potential application in quantum data storage. In this paper, we use classical LRCs to investigate quantum LRCs. We prove that the parameters of quantum LRCs are bounded by their classical counterparts. We deduce bounds on the parameters of quantum LRCs from bounds on the parameters of the classical ones. We establish a characterization of optimal pure quantum LRCs based on classical codes with specific properties. Using well-crafted classical LRCs as ingredients in the construction of quantum CSS codes, we offer the first construction of several families of optimal pure quantum LRCs.

arXiv:2312.11115IEEE Transactions on Information TheoryJan 24, 202521 citations

Asymmetric Quantum Codes

Giuliano G. La Guardia

Quantum science and technologyJan 1, 20200 citations

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}

\{r_X,r_Z\}

, in which

X

-type and

Z

-type errors are corrected from recovery sets of size at most

r_X+1

and

r_Z+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

\mathcal C_1

and

\mathcal C_2

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.

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.