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 · Quantum Information Processing · 2024
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.