The Clifford+$T$ quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group $langle H, Trangle$. The matrix $T$ can be considered the fourth root of Pauli $Z$, since $T^4 = Z$ or also the eig
hth 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.
We investigate the subclass of reversible functions that are self-inverse and relate them to reversible circuits that are equal to their reverse circuit, which are called palindromic circuits. We precisely determine which self-inverse functions can b
e realized as a palindromic circuit. For those functions that cannot be realized as a palindromic circuit, we find alternative palindromic representations that require an extra circuit line or quantum gates in their construction. Our analyses make use of involutions in the symmetric group $S_{2^n}$ which are isomorphic to self-inverse reversible function on $n$ variables.
