Toward Quantum CSS-T Codes from Sparse Matrices
Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Emily McMillon · 2024
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.