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

IEEE International Symposium on Information Theory (ISIT)

Stabilizer codes and Symbol-Pair Metric are Related

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

Published Jun 26, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1109/isit50566.2022.9834877

Related papers

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

A Symbol-Pair Decoder for CSS Codes

Vatsal Pramod Jha, Udaya Parampalli, Abhay Kumar Singh

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.

arXiv:2409.10979•Sep 17, 2024•0 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

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

Quantum XYZ Stabilizer Codes

Alessio Baldelli, Davide Orsucci, Francisco Lázaro +1 more

Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.

cs.ITCross-listedarXiv:2607.14988•Jul 17, 2026•0 citations

New construction of binary and nonbinary quantum stabilizer codes based on symmetric matrices

Duc Manh Nguyen, Sunghwan Kim

In this paper, we propose two construction methods for binary and nonbinary quantum stabilizer codes based on symmetric matrices. In the first construction, we use the identity and symmetric matrices to generate parity-check matrices that satisfy the symplectic inner product (SIP) for the construction of quantum stabilizer codes. In the second construction, we modify the first construction to generate parity-check matrices based on the Calderbank–Shor–Stean structure for the construction of quantum stabilizer codes. The binary and nonbinary quantum stabilizer codes whose parameters achieve equality of the quantum singleton bound are investigated with the code lengths ranging from 4 to 12.

International Journal of Modern Physics B•Sep 30, 2019•7 citations
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]].
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.
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.
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]].
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
In this paper, we propose two construction methods for binary and nonbinary quantum stabilizer codes based on symmetric matrices. In the first construction, we use the identity and symmetric matrices to generate parity-check matrices that satisfy the symplectic inner product (SIP) for the construction of quantum stabilizer codes. In the second construction, we modify the first construction to generate parity-check matrices based on the Calderbank–Shor–Stean structure for the construction of quantum stabilizer codes. The binary and nonbinary quantum stabilizer codes whose parameters achieve equality of the quantum singleton bound are investigated with the code lengths ranging from 4 to 12.