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 〈H, T〉. 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.