It dawned on me today that when I refer to the single-qubit Pauli group there are several different groups to which I could be referring.

Take the group generated by the Pauli matrices: $\sigma_x$, $\sigma_y$, and $\sigma_z$. Multiplying these together in different combinations yields the Pauli matrices, together with the identity, multiplied by factors of $\pm1$ and $\pm i$. This is a group with 16 elements, and masquerades under the name $\mathrm{C}_4\circ \mathrm{D}_4$ on GroupNames, though happily Pauli group is given as one of its aliases.

This is very nice. However, it’s also useful to think of the Pauli group as a finite-dimensional version of the Heisenberg group. This group is generated by $\sigma_x$ and $\sigma_z$, and has only 8 elements: $\{\pm1,\pm\sigma_x,\pm\sigma_z,\pm i\sigma_y\}$. It’s sometimes notated $\mathrm{He}_2$, and is isomorphic to the Dihedral group $\mathrm{D}_4$.

In quantum mechanics, one usually thinks of the unitaries that act on a qubit as elements of the special unitary group $\mathrm{SU}(2)$. The way we’ve constructed the previous two Pauli groups as groups of matrices gives us many matrices with non unit determinant—for example, $\mathrm{det}(\sigma_x)=-1$. If we instead take the group generated by $i\sigma_x$ and $i\sigma_z$, we get the subgroup of $\mathrm{C}_4\circ\mathrm{D}_4$ with unit determinant, which is known as the quaternion group $\mathrm{Q}_8$, since Pauli matrices with $i$s in front of them behave just like the $i$, $j$, and $k$ elements of the quaternions.

But that’s not all! Even the special unitaries have some redundancy, since the element $-1$ only puts a phase on quantum states, which is a trivial action. If we want the group of actions the Pauli matrices perform on quantum states, we need to remove this trivial action to get a subgroup of the projective unitary group $\mathrm{PU}(2)$. This means identifying the matrices in the quaternion group that differ only by a minus sign (or, equivalently, identifying the matrices in $\mathrm{C}_4\circ\mathrm{D}_4$ that differ by a complex phase), and gives us an abelian group with four elements, known as the Klein four-group and notated several ways including $\mathrm{V}_4$ and $\mathrm{C}_2^2$

So, next time you hear someone mention the single-qubit Pauli group, make sure to ask them which single-qubit Pauli group they have in mind!