Transversal gates for quantum CSS codes

TL;DR

This paper characterizes the set of diagonal transversal gates for CSS codes, providing explicit equations and extending results to a broad family of codes including monomial, polar, and CSS-T codes.

Contribution

It introduces a method to compute transversal gates and stabilizers for CSS codes, extending previous results to new families like monomial and polar codes.

Findings

Explicit equations for transversal gate groups

Complete characterization for monomial CSS codes

Extension of results to CSS-T and polar codes

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.