Translating between the roots of the identity in quantum computers

TL;DR

This paper explores the mathematical structure of Pauli root groups in quantum computing, generalizing roots of Pauli matrices and translation matrices, and analyzing their properties and relations.

Contribution

It introduces a formal framework for Pauli root groups, studies their finiteness, and characterizes their subgroup and equality relations.

Findings

Identified conditions for finiteness and infiniteness of Pauli root groups

Established criteria for subgroup relations among these groups

Provided a formal classification of the groups' structural properties

Abstract

The Clifford+$T$ quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group $\langle H, T\rangle$. The matrix $T$ can be considered the fourth root of Pauli $Z$, since $T^4 = Z$ or also the eighth root of the identity $I$. The Hadamard matrix $H$ can be used to translate between the Pauli matrices, since $(HTH)^4$ gives Pauli $X$. We are generalizing both these roots of the Pauli matrices (or roots of the identity) and translation matrices to investigate the groups they generate: the so-called Pauli root groups. In this work we introduce a formalization of such groups, study finiteness and infiniteness properties, and precisely determine equality and subgroup relations.