Toward Quantum CSS-T Codes from Sparse Matrices

TL;DR

This work presents a hull-centric characterization of CSS-T codes, showing that a CSS-T pair $(C_1,C_2)$ is equivalent to $C_2 \subseteq \mathrm{Hull}(C_1) \cap \mathrm{Hull}(C_1^2)$, and connects this condition to the relative hull $\mathrm{Hull}_{C_1^2}(C_1)$. It establishes puncturing results: if $C_2$ is degenerated on a coordinate, puncturing preserves the CSS-T property, enabling iterative construction via coordinate removal. The authors propose a practical, computational pathway using quasi-cyclic LDPC/LDGM codes to realize sparse CSS-T codes, providing Magma code and toy examples that demonstrate how QC-LDPC/LDGM frameworks can yield small CSS-T instances and pave the way for scalable, sparse quantum CSS-T codes with transversal gates. Together, these results offer both theoretical insight and actionable methods toward finding LDPC/LDGM CSS-T code families with potential fault-tolerant advantages.

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.

Theorems & Definitions (25)

• Proposition 1
• proof
• Definition 2
• Theorem 3
• Corollary 4
• Definition 5
• Definition 6
• Lemma 7
• proof
• Theorem 8