Toward Quantum CSS-T Codes from Sparse Matrices

TL;DR

This paper characterizes CSS-T quantum error-correcting codes using properties of binary linear codes and proposes a computational approach with Magma code and quasi-cyclic codes to find quantum LDPC or LDGM CSS-T codes.

Contribution

It provides a new characterization of CSS-T codes via code hulls and introduces methods for constructing such codes using sparse matrices and quasi-cyclic codes.

Findings

CSS-T codes are characterized by hull intersections of binary codes.

Puncturing codes preserves the CSS-T property under certain conditions.

Magma code and quasi-cyclic codes are used to computationally search for quantum LDPC/LDGM CSS-T codes.

Abstract

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 $C_2 \subset \operatorname{Hull}(C_1) \cap \operatorname{Hull}(C_1^2)$, 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 $(C_1|_i,C_2|_i)$ 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.