Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2409.10979

A Symbol-Pair Decoder for CSS Codes

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

Published Sep 17, 2024• 0 citations
arXivHTMLPDF

Abstract

The relation between stabilizer codes and binary codes provided by Gottesman and Calderbank et al. is a celebrated result, as it allows the lifting of classical codes to quantum codes. An equivalent way to state this result is that the work allows us to lift decoders for classical codes over the Hamming metric to decoders for stabilizer quantum codes. A natural question to consider: Can we do something similar with decoders for classical codes considered over other metrics? i.e., Can we lift decoders for classical codes over other metrics to obtain decoders for stabilizer quantum codes? In our current work, we answer this question in the affirmative by considering classical codes over the symbol-pair metric. In particular, we present a relation between the symplectic weight and the symbol-pair weight and use it to improve the error correction capability of CSS-codes (a well-studied class of stabilizer codes) obtained from cyclic codes.

DOI
10.48550/arxiv.2409.10979

Related papers

Stabilizer codes and Symbol-Pair Metric are Related

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

In [3], the relation between stabilizer codes and binary codes over the symplectic inner product and symplectic weight was established. In the current work, we present a relation between the symplectic weight and symbol-pair weight and use it to construct stabilizer codes of length n from binary codes of length n defined over the symbol-pair metric and Euclidean inner product. In particular, we use certain length n binary LCD codes over the symbol-pair metric to obtain stabilizer codes of length n. We also present the Modified CSS construction which outperforms the CSS construction in the given setup.

IEEE International Symposium on Information Theory (ISIT)•Jun 26, 2022•0 citations

Quantum codes via binary codes over symbol-pair metric

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

Stabilizer codes, introduced in \cite{gottesman}, \cite{GF(4)}, have been a prominent example of quantum codes constructed via classical codes. The paper \cite{GF(4)}, introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we present a few stabilizer code constructions considering binary codes over the symbol-pair metric (see \cite{sp}). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters $[[15,$ $4,$ $3\leq d\leq7]]$, $[[31,$ $10,$ $5\leq d\leq9]]$, $[[62,$ $20,$ $5\leq d]]$, $[[63,$ $18,$ $7\leq d]]$.

•Sep 2, 2021•0 citations

Quasi-cyclic low-density parity-check stabilizer codes

Feng Shi, Zhiyuan Yan

Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.

IEEE Military Communications Conference (MILCOM)•Oct 1, 2012•0 citations

Quantum codes via binary codes over symbol-pair metric

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

Stabilizer codes, introduced in \cite{gottesman}, \cite{GF(4)}, have been a prominent example of quantum codes constructed via classical codes. The paper \cite{GF(4)}, introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we present a few stabilizer code constructions considering binary codes over the symbol-pair metric (see \cite{sp}). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters $[[15,$ $4,$ $3\leq d\leq7]]$, $[[31,$ $10,$ $5\leq d\leq9]]$, $[[62,$ $20,$ $5\leq d]]$, $[[63,$ $18,$ $7\leq d]]$.

•Feb 3, 2022•1 citation

Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

Yansheng Wu, Yoonjin Lee

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

Mathematics•Sep 3, 2020•8 citations

Reinterpreting the Stabilizer Formalism

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.

•Sep 1, 2021•0 citations
The relation between stabilizer codes and binary codes provided by Gottesman and Calderbank et al. is a celebrated result, as it allows the lifting of classical codes to quantum codes. An equivalent way to state this result is that the work allows us to lift decoders for classical codes over the Hamming metric to decoders for stabilizer quantum codes. A natural question to consider: Can we do something similar with decoders for classical codes considered over other metrics? i.e., Can we lift decoders for classical codes over other metrics to obtain decoders for stabilizer quantum codes? In our current work, we answer this question in the affirmative by considering classical codes over the symbol-pair metric. In particular, we present a relation between the symplectic weight and the symbol-pair weight and use it to improve the error correction capability of CSS-codes (a well-studied class of stabilizer codes) obtained from cyclic codes.
In [3], the relation between stabilizer codes and binary codes over the symplectic inner product and symplectic weight was established. In the current work, we present a relation between the symplectic weight and symbol-pair weight and use it to construct stabilizer codes of length n from binary codes of length n defined over the symbol-pair metric and Euclidean inner product. In particular, we use certain length n binary LCD codes over the symbol-pair metric to obtain stabilizer codes of length n. We also present the Modified CSS construction which outperforms the CSS construction in the given setup.
Stabilizer codes, introduced in \cite{gottesman}, \cite{GF(4)}, have been a prominent example of quantum codes constructed via classical codes. The paper \cite{GF(4)}, introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we present a few stabilizer code constructions considering binary codes over the symbol-pair metric (see \cite{sp}). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters [[15,[[15,[[15, 4,4,4, 3≤d≤7]]3\leq d\leq7]]3≤d≤7]], [[31,[[31,[[31, 10,10,10, 5≤d≤9]]5\leq d\leq9]]5≤d≤9]], [[62,[[62,[[62, 20,20,20, 5≤d]]5\leq d]]5≤d]], [[63,[[63,[[63, 18,18,18, 7≤d]]7\leq d]]7≤d]].
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
Stabilizer codes, introduced in \cite{gottesman}, \cite{GF(4)}, have been a prominent example of quantum codes constructed via classical codes. The paper \cite{GF(4)}, introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we present a few stabilizer code constructions considering binary codes over the symbol-pair metric (see \cite{sp}). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters [[15,[[15,[[15, 4,4,4, 3≤d≤7]]3\leq d\leq7]]3≤d≤7]], [[31,[[31,[[31, 10,10,10, 5≤d≤9]]5\leq d\leq9]]5≤d≤9]], [[62,[[62,[[62, 20,20,20, 5≤d]]5\leq d]]5≤d]], [[63,[[63,[[63, 18,18,18, 7≤d]]7\leq d]]7≤d]].
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.