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 Transactions on Quantum Engineering

$\mathbb {F}_p$-Linear and $\mathbb {F}_{p^m}$-Linear Qudit Codes From Dual-Containing Classical Codes

Priya J. Nadkarni, Shayan Srinivasa Garani

Published Jan 1, 2021• 4 citations
Publisher

Abstract

Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field Fpm(p is prime and m is an integer) satisfying the dual containing criteria D1⊥⊂ D2using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the Fp-linear CSS codes and the well-known Fpm-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb Fp-linear and Fpm-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For Fpm-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances dxand dzare equal to d2and d1, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D1and D2, where one code is an Fpl-linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the Fpm-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.

DOI
10.1109/tqe.2021.3078152

Related papers

Duality of Quantum and Classical Error Correction Codes: Design Principles and Examples

Zunaira Babar, Daryus Chandra, Hung Viet Nguyen +4 more

Quantum error correction codes (QECCs) can be constructed from the known classical coding paradigm by exploiting the inherent isomorphism between the classical and quantum regimes, while also addressing the challenges imposed by the strange laws of quantum physics. In this spirit, this paper provides deep insights into the duality of quantum and classical coding theory, hence aiming for bridging the gap between them. Explicitly, we survey the rich history of both classical as well as quantum codes. We then provide a comprehensive slow-paced tutorial for constructing stabilizer-based QECCs from arbitrary binary as well as quaternary codes, as exemplified by the dual-containing and non-dual-containing Calderbank-Shor-Steane (CSS) codes, non-CSS codes and entanglement-assisted codes. Finally, we apply our discussions to two popular code families, namely to the family of Bose-Chaudhuri-Hocquenghem as well as of convolutional codes and provide detailed design examples for both their classical as well as their quantum versions.

IEEE Communications Surveys & Tutorials•Jan 1, 2019•65 citations

Quantum AMDS Negacyclic Codes of Length 2 p s Over F p m

Hai Quang Dinh, Bac Trong Nguyen, Nui V Nguyen +2 more

In this article, we investigate all almost maximum distance separable (briefly, AMDS) negacyclic codes of length 2psover the finite field Fpm, wherepis an odd prime. A necessary and sufficient condition for a code to be an AMDS and NMDS negacyclic code of length 2psover Fpmis also provided. In addition, we construct quantum AMDS (qAMDS) codes using both the Calderbank-Shor-Steane (CSS) and Hermitian constructions. These newly constructed codes have different parameters compared to previous constructions. Moreover, our qAMDS codes are better than all known quantum error correction (QEC) codes.

IEEE Access•Aug 4, 2025•0 citations

Classical product code constructions for quantum Calderbank-Shor-Steane codes

Dimiter Ostrev, Davide Orsucci, Francisco Lázaro +1 more

Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3 -fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.

arXiv:2209.13474•Quantum•Jul 22, 2024•8 citations

Calderbank-Shor-Steane codes on group-valued qudits

Ben T. McDonough, Jian-Hao Zhang, Victor V. Albert +1 more

Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting $X$- and $Z$-type checks. We introduce CSS-like codes on $G$-valued qudits for any finite group $G$ that reduce to qubit CSS codes for $G = \mathbb{Z}_2$ yet generalize the Kitaev quantum double model for general groups. The $X$-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while $Z$-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when $G$ is non-Abelian and simple, every $G$-covariant group-CSS code with suitably upper-bounded $Z$-check weight and lower-bounded $Z$-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.

quant-pharXiv:2602.19558•Feb 24, 2026•0 citations

Quantum error correcting codes based on privacy amplification

Zhicheng Luo

Calderbank-Shor-Steane (CSS) quantum error-correcting codes are based on pairs of classical codes which are mutually dual containing. Explicit constructions of such codes for large blocklengths and with good error correcting properties are not easy to find. In this paper we propose a construction of CSS codes which combines a classical code with a two-universal hash function. We show, using the results of Renner and Koenig, that the communication rates of such codes approach the hashing bound on tensor powers of Pauli channels in the limit of large block-length. While the bit-flip errors can be decoded as efficiently as the classical code used, the problem of efficiently decoding the phase-flip errors remains open.

arXiv:0808.1392•Aug 10, 2008•3 citations

Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over $\mathbb {F}_{p^k}$

Priya J. Nadkarni, Shayan Srinivasa Garani

We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over Fq, where q = pk, p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over Fq. We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.

IEEE Transactions on Quantum Engineering•Jan 1, 2020•10 citations
Fp\mathbb {F}_pFp​-Linear and Fpm\mathbb {F}_{p^m}Fpm​-Linear Qudit Codes From Dual-Containing Classical Codes
Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field Fpm(p is prime and m is an integer) satisfying the dual containing criteria D1⊥⊂ D2using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the Fp-linear CSS codes and the well-known Fpm-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb Fp-linear and Fpm-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For Fpm-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances dxand dzare equal to d2and d1, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D1and D2, where one code is an Fpl-linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the Fpm-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.
Quantum error correction codes (QECCs) can be constructed from the known classical coding paradigm by exploiting the inherent isomorphism between the classical and quantum regimes, while also addressing the challenges imposed by the strange laws of quantum physics. In this spirit, this paper provides deep insights into the duality of quantum and classical coding theory, hence aiming for bridging the gap between them. Explicitly, we survey the rich history of both classical as well as quantum codes. We then provide a comprehensive slow-paced tutorial for constructing stabilizer-based QECCs from arbitrary binary as well as quaternary codes, as exemplified by the dual-containing and non-dual-containing Calderbank-Shor-Steane (CSS) codes, non-CSS codes and entanglement-assisted codes. Finally, we apply our discussions to two popular code families, namely to the family of Bose-Chaudhuri-Hocquenghem as well as of convolutional codes and provide detailed design examples for both their classical as well as their quantum versions.
In this article, we investigate all almost maximum distance separable (briefly, AMDS) negacyclic codes of length 2psover the finite field Fpm, wherepis an odd prime. A necessary and sufficient condition for a code to be an AMDS and NMDS negacyclic code of length 2psover Fpmis also provided. In addition, we construct quantum AMDS (qAMDS) codes using both the Calderbank-Shor-Steane (CSS) and Hermitian constructions. These newly constructed codes have different parameters compared to previous constructions. Moreover, our qAMDS codes are better than all known quantum error correction (QEC) codes.
Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both X parity checks and Z parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC 3 -fold product CSS code having parameters [[512,174,8]] are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.
Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting XXX- and ZZZ-type checks. We introduce CSS-like codes on GGG-valued qudits for any finite group GGG that reduce to qubit CSS codes for G=Z2G = \mathbb{Z}_2G=Z2​ yet generalize the Kitaev quantum double model for general groups. The XXX-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while ZZZ-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when GGG is non-Abelian and simple, every GGG-covariant group-CSS code with suitably upper-bounded ZZZ-check weight and lower-bounded ZZZ-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.
Calderbank-Shor-Steane (CSS) quantum error-correcting codes are based on pairs of classical codes which are mutually dual containing. Explicit constructions of such codes for large blocklengths and with good error correcting properties are not easy to find. In this paper we propose a construction of CSS codes which combines a classical code with a two-universal hash function. We show, using the results of Renner and Koenig, that the communication rates of such codes approach the hashing bound on tensor powers of Pauli channels in the limit of large block-length. While the bit-flip errors can be decoded as efficiently as the classical code used, the problem of efficiently decoding the phase-flip errors remains open.
Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over Fpk\mathbb {F}_{p^k}Fpk​
We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over Fq, where q = pk, p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over Fq. We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.