A Classification of Transversal Clifford Gates for Qubit Stabilizer Codes

TL;DR

This paper classifies stabilizer codes based on the diagonal Clifford gates implementable transversally, revealing six distinct families of such gate groups and advancing the theoretical understanding of code symmetries.

Contribution

It introduces a complete classification of diagonal Clifford symmetries for stabilizer codes using matrix algebra methods, expanding the theoretical framework.

Findings

Six distinct families of diagonal transversal Clifford gate groups identified

Complete classification of diagonal Clifford symmetries for stabilizer codes provided

Theoretical framework based on matrix algebras of endomorphisms developed

Abstract

This work classifies stabilizer codes by the set of diagonal Clifford gates that can be implemented transversally on them. We show that, for any stabilizer code, its group of diagonal transversal Clifford gates on $\ell$ code blocks must be one of six distinct families of matrix groups. We further develop the theory of classifying stabilizer codes by via matrix algebras of endomorphisms first introduced by Rains, and give a complete classification of the diagonal Clifford symmetries of $\ell$ code blocks. A number of corollaries are given in the final section.