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Transversal gates for quantum CSS codes

Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Narayanan Rengaswamy, Rodrigo San-José

TL;DR

The paper tackles the problem of identifying diagonal transversal gates that fix a CSS code and determining their induced logical actions. It develops a group-theoretic framework with $H_N(Q)$, $T_N(Q)$, and $Id_N(Q)$, and shows how diagonal gates fixing a CSS code reduce to the case $Q' = Q(C_1,C_2)$ via a basis-preserving conjugation, enabling a systematic analysis. For CSS codes arising from monomial codes, it provides a complete, constructive description of the transversal gates and their actions, proving that the gate set is entirely determined by the code and is independent of basis choices; it also connects these results to CSS-T, triorthogonal, and divisible codes, and describes explicit logical actions using controlled $U(a)$ gates. The work supports fault-tolerant quantum computation by clarifying when and how transversal diagonal gates can realize logical operations or identities, and suggests extensions to broader code families and potential impacts on magic-state distillation and universality.

Abstract

In this paper, we focus on the problem of computing the set of diagonal transversal gates fixing a CSS code. We determine the logical actions of the gates as well as the groups of transversal gates that induce non-trivial logical gates and logical identities. We explicitly declare the set of equations defining the groups, a key advantage and differentiator of our approach. We compute the complete set of transversal stabilizers and transversal gates for any CSS code arising from monomial codes, a family that includes decreasing monomial codes and polar codes. As a consequence, we recover and extend some results in the literature on CSS-T codes, triorthogonal codes, and divisible codes.

Transversal gates for quantum CSS codes

TL;DR

The paper tackles the problem of identifying diagonal transversal gates that fix a CSS code and determining their induced logical actions. It develops a group-theoretic framework with , , and , and shows how diagonal gates fixing a CSS code reduce to the case via a basis-preserving conjugation, enabling a systematic analysis. For CSS codes arising from monomial codes, it provides a complete, constructive description of the transversal gates and their actions, proving that the gate set is entirely determined by the code and is independent of basis choices; it also connects these results to CSS-T, triorthogonal, and divisible codes, and describes explicit logical actions using controlled gates. The work supports fault-tolerant quantum computation by clarifying when and how transversal diagonal gates can realize logical operations or identities, and suggests extensions to broader code families and potential impacts on magic-state distillation and universality.

Abstract

In this paper, we focus on the problem of computing the set of diagonal transversal gates fixing a CSS code. We determine the logical actions of the gates as well as the groups of transversal gates that induce non-trivial logical gates and logical identities. We explicitly declare the set of equations defining the groups, a key advantage and differentiator of our approach. We compute the complete set of transversal stabilizers and transversal gates for any CSS code arising from monomial codes, a family that includes decreasing monomial codes and polar codes. As a consequence, we recover and extend some results in the literature on CSS-T codes, triorthogonal codes, and divisible codes.
Paper Structure (5 sections, 20 theorems, 63 equations)

This paper contains 5 sections, 20 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Transversal operators for CSS codes with arbitrary characters
  3. Diagonal transversal operators for CSS codes
  4. Conclusion
  5. Acknowledgements

Key Result

Lemma 2.2

Let $C_2\subseteq C_1\subseteq\mathbb{F}_2^n$, $y_x,y_z\in\mathbb{F}_2^n$, and $L\subseteq\mathbb{F}_2^n$ a set of representatives of $C_1/C_2$. Then, the set $\{|C_2+w+y_z\rangle\ :\ w\in L\}$ is an orthonormal basis for $Q=Q(C_1,C_2,y_x,y_z)$, where Moreover, if $d_x=\min\{wt(w)\ :\ w\in C_1\setminus C_2\}$, $d_z=\min\{wt(w)\ :\ w\in C_2^\perp\setminus C_1^\perp\}$, $k_1=\dim C_1$, and $k_2=\di

Theorems & Definitions (47)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 37 more