Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$

Hai Quang Dinh, Bac Trong Nguyen, Hiep L. Thi +2 more

In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.

EPJ Quantum TechnologyJun 17, 20260 citations

New constructions of entanglement-assisted quantum codes

Adel N. Alahmadi, Ahmad N. Al‐Kenani, Rola Hijazi +3 more

Cryptography and CommunicationsJun 26, 202125 citations

Asymmetric Entanglement-Assisted Quantum MDS Codes Constructed from Constacyclic Codes

Jian‐Zhang Chen, Wanchuan Fang, Shuo Zhou +5 more

Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry.Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry.

EntropyNov 30, 202310 citations

Dualities and identities for entanglement-assisted quantum codes

Ching-Yi Lai, Todd A. Brun, Mark M. Wilde

arXiv:1010.5506Quantum Information ProcessingDec 16, 201337 citations

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

Carlos Galindo, Fernando Hernando, Ryutaroh Matsumoto +1 more

The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.

arXiv:1911.10031IEEE AccessJan 1, 202028 citations

Asymmetric Entanglement-Assisted Quantum Locally Recoverable Codes

Vijay Kumar, Ramakrishna Bandi

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

Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits

p^{s}

over

\mathbb{F}_{p^{m}}

In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits

p^{s}

over

\mathbb{F}_{p^{m}}

. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits

p^{s}

over

\mathbb{F}_{p^{m}}

. When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.

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.