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:2209.13474 · Quantum

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

Dimiter Ostrev, Davide Orsucci, Francisco Lázaro, Balázs Matuz

Published Jul 22, 2024• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2024-07-22-1420

Related papers

Quantum CSS LDPC Codes based on Dyadic Matrices for Belief Propagation-based Decoding

Alessio Baldelli, Massimo Battaglioni, Jonathan Mandelbaum +2 more

Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.

cs.ITCross-listedarXiv:2601.08636•IEEE International Symposium on Information Theory (ISIT)•Jan 14, 2026•0 citations

Neural Belief Propagation Decoders for Concatenated Calderbank-Shor-Steane Codes

Jihao Fan, Jinbing Zhang

Quantum low-density parity check (QLDPC) codes have rapidly developed in the field of quantum error correction in recent years. A variety of quantum code families have emerged from numerous research activities. In this paper, we employ concatenated Calderbank-Shor-Steane (CSS) codes, hypergraph product codes, and Bacon-Shor codes to train belief-propagation (BP) and neural BP (NBP) decoders. At the same time, we employ a concatenation scheme in BP and NBP, the amplified construction of CSS codes is realized, and the training of longer length parameter quantum codes is realized. The numerical results show that the concatenated CSS codes demonstrate better error correction performance compared to both hypergraph product codes and Bacon-Shor codes. In addition, the concatenated CSS codes are trained using the concatenation scheme to improve the error correction performance of the decoder.

International Conference on Communications•Jun 9, 2024•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

High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures

Koki Okada, Kenta Kasai

We construct a quantum low-density parity-check code family from a length-$512$ Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are $(3,8)$-regular with girth $8$. The base code has parameters $[[512,174,8]]$. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor $P=32$, which has parameters $[[16384,4142,18\le d\le32]]$, under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most $16$ on both sides, giving $d_X,d_Z\ge18$. Explicit weight-$32$ non-stabilizer logical representatives give the upper bounds $d_X,d_Z\le32$. A belief-propagation decoder with post-processing achieved frame error rate about $10^{-8}$ at $p=0.085$; an independently observed logical residual of weight $40$ is consistent with the sharper structural bound.

quant-pharXiv:2604.20838•Apr 23, 2026•0 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

High-dimensional quantum X Y Z product codes for biased noise

Zhipeng Liang, Zhengzhong Yi, Fusheng Yang +3 more

The three-dimensional (3D) quantum XYZ product can construct a class of non-Calderbank-Shor-Stearne (non-CSS) quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance so far and whether this code construction can be generalized to higher dimension is an open question. In this paper, we first study the error-correcting performance of the 3D Chamon code, which is an instance of the 3D XYZ product of three repetition codes. Second, we show that the 3D XYZ product can be generalized to four dimensions and propose four-dimensional (4D) XYZ product code construction, which constructs a class of non-CSS quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, which we refer to as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli-$Z$-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.

Physical Review A•Nov 19, 2025•0 citations
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.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Quantum low-density parity check (QLDPC) codes have rapidly developed in the field of quantum error correction in recent years. A variety of quantum code families have emerged from numerous research activities. In this paper, we employ concatenated Calderbank-Shor-Steane (CSS) codes, hypergraph product codes, and Bacon-Shor codes to train belief-propagation (BP) and neural BP (NBP) decoders. At the same time, we employ a concatenation scheme in BP and NBP, the amplified construction of CSS codes is realized, and the training of longer length parameter quantum codes is realized. The numerical results show that the concatenated CSS codes demonstrate better error correction performance compared to both hypergraph product codes and Bacon-Shor codes. In addition, the concatenated CSS codes are trained using the concatenation scheme to improve the error correction performance of the decoder.
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.
We construct a quantum low-density parity-check code family from a length-512512512 Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are (3,8)(3,8)(3,8)-regular with girth 888. The base code has parameters [[512,174,8]][[512,174,8]][[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32P=32P=32, which has parameters [[16384,4142,18≤d≤32]][[16384,4142,18\le d\le32]][[16384,4142,18≤d≤32]], under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18. Explicit weight-323232 non-stabilizer logical representatives give the upper bounds dX,dZ≤32d_X,d_Z\le32dX​,dZ​≤32. A belief-propagation decoder with post-processing achieved frame error rate about 10−810^{-8}10−8 at p=0.085p=0.085p=0.085; an independently observed logical residual of weight 404040 is consistent with the sharper structural bound.
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.
The three-dimensional (3D) quantum XYZ product can construct a class of non-Calderbank-Shor-Stearne (non-CSS) quantum codes by using three classical codes. However, there has been limited study on their error-correcting performance so far and whether this code construction can be generalized to higher dimension is an open question. In this paper, we first study the error-correcting performance of the 3D Chamon code, which is an instance of the 3D XYZ product of three repetition codes. Second, we show that the 3D XYZ product can be generalized to four dimensions and propose four-dimensional (4D) XYZ product code construction, which constructs a class of non-CSS quantum codes by using either four classical codes or two CSS quantum codes. Compared with the 4D homological product, we show that the 4D XYZ product can construct non-CSS codes with higher code dimension or code distance. Third, we consider two instances of the 4D XYZ product, which we refer to as the 4D Chamon code and the 4D XYZ product concatenated code, respectively. Our simulation results show that the 4D XYZ product can construct non-CSS codes with better error-correcting performance against Pauli-ZZZ-biased noise than CSS codes constructed by the 4D homological product. Finally, we present the geometric arrangement of the 4D Chamon code within a 4D cubic lattice, demonstrating that it possesses two key characteristics of fracton models, which strongly suggest that it is a novel 4D fracton model.