# What are the elements of quotienting the Pauli group $\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$, and how to do calculations with it?

Let $$\widetilde{\mathcal{P}}_n = \langle X_1,X_2,\dots,X_n,Z_1,\dots,Z_n\rangle$$ together with all the phases $$\{\pm 1, \pm i\}$$ the regular Pauli group, and $$N = \langle \pm i I\rangle$$. I would like to understand better the group theoretic construction:

$$\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$$

In this work, arXiv, the authors mention that this group corresponds to the Pauli operators with only $$+1$$ phase. However, in group theoretic terms, this construction generates sets as elemens instead of Pauli operators. In effect, the elements of $$G/N$$ in group theoretic 'jargon' should be left/right cosets.

However, the authors proceed to calculate, for instance $$\sum_{P \in \mathcal{P}_n} \langle \psi \vert P \vert \psi \rangle$$ for any $$\vert \psi \rangle$$. How can I understand the construction of the set $$\mathcal{P}_n$$, if possible, in general and not only for this specific case?

• You are correct, and the authors are using slightly lose language. This is instructive en.wikipedia.org/wiki/Quotient_group In short, we don't care about the phase of the Pauli operators, as any such phase has no impact on measurement probabilities. So we create an equivalence class of all Pauli operators that differ by a phase. It is not very different from working with the modular group n=12 (clock time), where anytime we get a number bigger than 12, we consider it equivalent to time from 0 to 11. Feb 8 at 23:00

Succinctly, what you call the $$n$$-qubit Pauli group $$\tilde{\mathcal{P}}_n$$ is a non-abelian group with $$4^{n+1}$$ elements. As you explained, you can create the normal group $$N:= \left< \pm i I \right> = \{ I, -I, iI, -iI \}.$$
Next, what you denote $$\mathcal{P}_n$$ is defined as the quotient group: $$\mathcal{P}_n := \tilde{\mathcal{P}}_n / N\,.$$ This is what the authors in the reference above call the abelian $$n$$-qubit Pauli group. Formally, the associated canonical quotient map is denoted by $$\pi$$, i.e., $$\pi: \tilde{\mathcal{P}}_n \rightarrow \mathcal{P}_n$$, which is surjective and is given by $$\pi(g) = gN$$.
Informally, we note that $$\pi(g) = \pi(-g) = \pi(ig) = \pi(-ig)$$, which is why there are only $$4^n$$ (different sets of) elements in the quotient group which take the form: $$\{ p, -p, ip, -ip \}$$, with $$p$$ a Pauli operator with $$+1$$ phase. (The operators $$-p$$, $$ip$$, and $$-ip$$ will give rise exactly to the same set.)
Each of these elements can be regarded as an equivalent class, denoted $$\left[ p \right]$$. It is in this sense that when the authors in your reference write: $$\sum_{p\in \mathcal{P}_n}$$ they mean that $$p$$ is a Pauli which has phase $$+1$$. (The Paulis $$-p$$, $$ip$$, $$-ip$$ belong to the same equivalence class as $$p$$.)