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:2312.17518 · Quantum Information Processing

An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance

Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Diego Ruano, Rodrigo San–José, Ivan Soprunov

Published Jun 6, 2024• 12 citations
arXivHTMLPDF

Abstract

Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $$(C_1, C_2)$$ ( C 1 , C 2 ) such that $$C_1$$ C 1 contains $$C_2$$ C 2 , $$C_2$$ C 2 is even, and the shortening of the dual of $$C_1$$ C 1 with respect to the support of each codeword of $$C_2$$ C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $$(C_1, C_2)$$ ( C 1 , C 2 ) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.

DOI
10.1007/s11128-024-04427-5

Related papers

Quantum Matrix-Product Codes: CSS-T Characterization and Maximality

Delio Jaramillo-Velez, Alessandro Neri, Adway Patra +2 more

CSS-T codes are quantum error-correcting codes that play an important role in fault-tolerant quantum computation, as they help mitigate the proliferation of errors. They admit a transversal T-gate and are defined from a pair of nested classical binary linear codes satisfying a specific algebraic condition expressed in terms of their Schur square. We extend this algebraic characterization to the propagation rule known as the $(u \mid u+v)$-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.

arXiv:2609.08520•Sep 8, 2026•0 citations

A New Family of Fault Tolerant Quantum Reed-Muller Codes

Harrison Eggers

Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.

•Jan 1, 2020•0 citations

Toward Quantum CSS-T Codes from Sparse Matrices

Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews +1 more

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair ($C_{1}, C_{2}$) of binary linear codes$C_{1}$and$C_{2}$that satisfy certain conditions. We prove that$C_{1}$and$C_{2}$form a CSS-T pair if and only if$C_{2}\subset \text{Hull}(C_{1})\cap \text{Hull}(C_{1}^{2})$, where the hull of a code is the intersection of the code with its dual. We show that if ($C_{1}, C_{2}$) is a CSS-T pair, and the code$C_{2}$is degenerated on$\{i\}$, meaning that the$i^{th}$-entry is zero for all the elements in$C_{2}$, then the pair of punctured codes ($C_{1}\vert_{i}, C_{2}\vert_{i}$) is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.

arXiv:2406.00425•Jul 7, 2024•4 citations

CSS-$T$ codes over Binary Extension Fields and their Physical Foundations

J. J. Postema, Fabrizio Conca, Alberto Ravagnani

We investigate the class of CSS-$T$ codes, a family of quantum error-correcting codes that allows for a transversal $T$-gate. We extend the definition of a pair of linear codes $(C_1,C_2)$, $C_i\subseteq\mathbb{F}_q^n$, forming a $q$-ary CSS-$T$ code over binary extension fields, and demonstrate the existence of asymptotically good sequences of LDPC CSS-$T$ codes over any such field.

arXiv:2507.17611•Jul 23, 2025•0 citations

Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly

K. Sai Mineesh Reddy, Navin Kashyap

This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.

quant-pharXiv:2601.08568•Jan 14, 2026•0 citations

Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS Codes

K. Sai Mineesh Reddy, Navin Kashyap

Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled $Z$-rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled $Z$-rotations.

quant-pharXiv:2608.19094•Aug 20, 2026•0 citations
Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C1,C2)(C_1, C_2)(C1​,C2​) ( C 1 , C 2 ) such that C1C_1C1​ C 1 contains C2C_2C2​ C 2 , C2C_2C2​ C 2 is even, and the shortening of the dual of C1C_1C1​ C 1 with respect to the support of each codeword of C2C_2C2​ C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C1,C2)(C_1, C_2)(C1​,C2​) ( C 1 , C 2 ) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.
CSS-T codes are quantum error-correcting codes that play an important role in fault-tolerant quantum computation, as they help mitigate the proliferation of errors. They admit a transversal T-gate and are defined from a pair of nested classical binary linear codes satisfying a specific algebraic condition expressed in terms of their Schur square. We extend this algebraic characterization to the propagation rule known as the (u∣u+v)(u \mid u+v)(u∣u+v)-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.
Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.
CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a pair (C1,C2C_{1}, C_{2}C1​,C2​) of binary linear codesC1C_{1}C1​andC2C_{2}C2​that satisfy certain conditions. We prove thatC1C_{1}C1​andC2C_{2}C2​form a CSS-T pair if and only ifC2⊂Hull(C1)∩Hull(C12)C_{2}\subset \text{Hull}(C_{1})\cap \text{Hull}(C_{1}^{2})C2​⊂Hull(C1​)∩Hull(C12​), where the hull of a code is the intersection of the code with its dual. We show that if (C1,C2C_{1}, C_{2}C1​,C2​) is a CSS-T pair, and the codeC2C_{2}C2​is degenerated on{i}\{i\}{i}, meaning that theithi^{th}ith-entry is zero for all the elements inC2C_{2}C2​, then the pair of punctured codes (C1∣i,C2∣iC_{1}\vert_{i}, C_{2}\vert_{i}C1​∣i​,C2​∣i​) is also a CSS-T pair. Finally, we provide Magma code based on our results and quasi-cyclic codes as a step toward finding quantum LDPC or LDGM CSS-T codes computationally.
CSS-TTT codes over Binary Extension Fields and their Physical Foundations
We investigate the class of CSS-TTT codes, a family of quantum error-correcting codes that allows for a transversal TTT-gate. We extend the definition of a pair of linear codes (C1,C2)(C_1,C_2)(C1​,C2​), Ci⊆FqnC_i\subseteq\mathbb{F}_q^nCi​⊆Fqn​, forming a qqq-ary CSS-TTT code over binary extension fields, and demonstrate the existence of asymptotically good sequences of LDPC CSS-TTT codes over any such field.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal ZZZ-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal TTT realizes the logical transversal S†S^{\dagger}S†; (b) show that the condition C2∗C1⊆C1⊥C_2 \ast C_1 \subseteq C_1^{\perp}C2​∗C1​⊆C1⊥​ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal TTT implements the logical identity and the logical transversal TTT, respectively.
Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆C1​, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled ZZZ-rotations.