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arXiv:2610.23889

Asymmetric Entanglement-Assisted Quantum Locally Recoverable Codes

Vijay Kumar, Ramakrishna Bandi

quant-phcs.ITSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

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.

Related papers

Entanglement-Assisted Quantum Locally Recoverable Codes: Bounds, Optimal Constructions, and Achievability

Vijay Kumar, Ramakrishna Bandi

This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.

arXiv:2608.06854•Aug 7, 2026•0 citations

Entanglement-Assisted Quantum Locally Recoverable Codes: Characterizations, Bounds, and Constructions

Yang Li, San Ling, Zhenliang Lu +2 more

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 $r$ 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 $\ell$-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.

arXiv:2607.27091•Jul 29, 2026•0 citations

CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

Evagoras Stylianou, Vinayak Ramkumar, Holger Boche +1 more

In this work, we study $(r,t,x)$ quantum locally recoverable codes (qLRCs) with locality $r$, $t$ recovery sets per qudit, and intersection parameter $x$. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an $(r,t,x)$-qLRC if and only if the underlying classical codes are $(r,t,x)$ classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing $(r,t,x)$-cLRCs, which yield binary $(r,t,x)$-qLRCs via the CSS construction. For CSS $(r,t,x)$-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.

cs.ITCross-listedarXiv:2608.10912•Aug 12, 2026•0 citations

Quantum Locally Recoverable Codes

Louis Golowich, Venkatesan Guruswami

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.

arXiv:2311.08653•Nov 15, 2023•1 citation

Alphabet-Dependent Bounds for Pure Quantum $(r,ρ)$-Locally Recoverable Codes

Vijay Kumar, Ramakrishna Bandi

A quantum $(r,ρ)$-locally recoverable code ($(r,ρ)$-qLRC) is a quantum code in which every qudit can be recovered from at most $r+ρ-1$ other qudits, even after $ρ-1$ additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure $(r,ρ)$-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.

quant-pharXiv:2608.28650•Aug 20, 2026•0 citations

Entanglement assisted quantum $(r,δ)$-locally recoverable codes

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz +1 more

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.

quant-pharXiv:2608.17118•Aug 19, 2026•0 citations
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.
This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case C1=C2=C\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}C1​=C2​=C, this reduces to a simple condition on the hull dimension s=dim⁡(C∩C⊥)s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)s=dim(C∩C⊥), which also fixes the entanglement count via c=n−k−sc=n-k-sc=n−k−s. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime k≤rk \le rk≤r, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size q>3q>3q>3 via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.
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.
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.
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.
Alphabet-Dependent Bounds for Pure Quantum (r,ρ)(r,ρ)(r,ρ)-Locally Recoverable Codes
A quantum (r,ρ)(r,ρ)(r,ρ)-locally recoverable code ((r,ρ)(r,ρ)(r,ρ)-qLRC) is a quantum code in which every qudit can be recovered from at most r+ρ−1r+ρ-1r+ρ−1 other qudits, even after ρ−1ρ-1ρ−1 additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure (r,ρ)(r,ρ)(r,ρ)-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.
Entanglement assisted quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes
Quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes are quantum error-correcting codes capable of correcting δ−1δ-1δ−1 qudit erasures within one subset of qudits of cardinality at most r+δ−1r+δ-1r+δ−1. In this paper, we introduce the more general framework of entanglement-assisted quantum (r,δ)(r,δ)(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)(r,δ)(r,δ)-local recoverability, and derive a Singleton-like bound. Furthermore, we construct optimal pure entan\-gle\-ment-assisted quantum (r,δ)(r,δ)(r,δ)-locally recoverable codes from several families of classical codes, including bivariate JJJ-affine variety codes, BCH codes, and homothetic-BCH codes.