Translating between the roots of the identity in quantum computers

The Clifford+T quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group . 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