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

Question

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?

Answer 1

It seems like the answer you are looking for may be found at the beginning of section 2 of this reference: https://arxiv.org/pdf/1909.08123.pdf.

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 $p$.)

To conclude, as explained by Abdullah Khalid, the phases have no influence on the overall expectation values that are being calculated, and so there is no point in summing over all Paulis with all phases. So the authors sum only over one Pauli from each equivalence class, and they choose that Pauli to be the one with positive phase wlog.