An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance

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 $(C_1,C_2)$ 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.

References in corpus (7)

• Magic state distillation with low overhead
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Classification of Small Triorthogonal Codes
• New Quantum Codes from CSS Codes
• Algebraic Quantum Codes: Linking Quantum Mechanics and Discrete Mathematics
• Classical Coding Problem from Transversal $T$ Gates
• Structure of CSS and CSS-T Quantum Codes

Cited by in corpus (2)

• Transversal Clifford and T-gate codes of short length and high distance
• CSS-T Codes from Reed Muller Codes