An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance
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arXiv
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| Format: | Preprint |
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2023
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| author | Camps-Moreno, Eduardo López, Hiram H. Matthews, Gretchen L. Ruano, Diego San-José, Rodrigo Soprunov, Ivan |
| author_facet | Camps-Moreno, Eduardo López, Hiram H. Matthews, Gretchen L. Ruano, Diego San-José, Rodrigo Soprunov, Ivan |
| contents | 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 $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $(C_1, C_2)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_17518 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance Camps-Moreno, Eduardo López, Hiram H. Matthews, Gretchen L. Ruano, Diego San-José, Rodrigo Soprunov, Ivan Information Theory 94B05 (Primary), 81P70, 11T71, 14G50 (Secondary) 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 $(C_1, C_2)$ such that $C_1$ contains $C_2$, $C_2$ is even, and the shortening of the dual of $C_1$ with respect to the support of each codeword of $C_2$ is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $(C_1, C_2)$ 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. |
| title | An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance |
| topic | Information Theory 94B05 (Primary), 81P70, 11T71, 14G50 (Secondary) |
| url | https://arxiv.org/abs/2312.17518 |