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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

TL;DR

This work provides a direct algebraic characterization of binary CSS-T codes by showing that a CSS-T pair $(C_1,C_2)$ satisfies $C_2\subset C_1\cap(C_1^{\star 2})^{\perp}$, with $C_2$ necessarily self-orthogonal. It introduces a poset of CSS-T pairs, identifies minimal and maximal elements, and proves a propagation rule that enlarges nondegenerate CSS-T codes while preserving the CSS-T property. The authors specialize the theory to cyclic and extended cyclic codes via cyclotomic cosets, giving a precise criterion $I_2\subset I_1$ and $n\notin(I_1+I_1+I_2)$ for CSS-T pairs, and show these can yield quantum codes with improved parameters over existing constructions. Applications to Reed–Muller, cyclic, and extended cyclic codes produce new families of quantum codes with competitive or superior parameters, demonstrating the practical impact for fault-tolerant quantum computation. The work also relates CSS-T codes to triorthogonal codes, highlighting trade-offs between transversal T gates and code parameters, and outlines open problems for extending the framework to $q$-ary codes and other classical families.

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.

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

TL;DR

This work provides a direct algebraic characterization of binary CSS-T codes by showing that a CSS-T pair satisfies , with necessarily self-orthogonal. It introduces a poset of CSS-T pairs, identifies minimal and maximal elements, and proves a propagation rule that enlarges nondegenerate CSS-T codes while preserving the CSS-T property. The authors specialize the theory to cyclic and extended cyclic codes via cyclotomic cosets, giving a precise criterion and for CSS-T pairs, and show these can yield quantum codes with improved parameters over existing constructions. Applications to Reed–Muller, cyclic, and extended cyclic codes produce new families of quantum codes with competitive or superior parameters, demonstrating the practical impact for fault-tolerant quantum computation. The work also relates CSS-T codes to triorthogonal codes, highlighting trade-offs between transversal T gates and code parameters, and outlines open problems for extending the framework to -ary codes and other classical families.

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 such that contains , is even, and the shortening of the dual of with respect to the support of each codeword of is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes 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.
Paper Structure (10 sections, 21 theorems, 57 equations, 2 tables)

This paper contains 10 sections, 21 theorems, 57 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of major results
  3. Motivating example
  4. Equivalent Definitions
  5. The poset of CSS-T pairs
  6. Cyclic codes
  7. Extended cyclic codes
  8. Relation to triorthogonal codes
  9. Conclusion
  10. Acknowledgements

Key Result

Theorem 2.1

Let $C_i\subset {\mathbb F}_{2}^n$ be linear codes of dimension $k_i$, for $i=1,2$, such that $C_2\subset C_1$. Then, there is an $[[n,k_1-k_2,d]]$ quantum code with

Theorems & Definitions (59)

  • Theorem 2.1: CSS Construction
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Corollary 3.1
  • proof
  • Definition 3.2
  • ...and 49 more