Is the Pauli group isomorphic to the Heisenberg group over a finite field?

Question

Let $ p $ be prime and let $ P_n(p) $ denote the Pauli group on $ n $ qudits each of size $ p $. Then $ P_n(p) $ and $ \text{Heis}_{2n+1}(\mathbb{F}_p) $ are both extraspecial $ p $ groups of order $ p^{2n+1} $. Are they isomorphic?

I think the answer is obviously yes using the symplectic representation of the Pauli group, but I don't see this fact explicitly stated anywhere so I was wondering if I'm missing something. Basically an arbitrary element of $ \text{Heis}_{2n+1}(\mathbb{F}_p) $ is given by $ (v,\zeta^i,w) $ where $ v,w \in \mathbb{F}_p^n $ and $ \zeta $ is a central element of order $ p $ (in the standard representation $ \zeta $ is just a primitive $ p $th root of unity). Then the bijection $ \text{Heis}_{2n+1}(\mathbb{F}_p) \to P_n(p) $ $$ (v,\zeta^i,w) \mapsto \zeta^i X^vZ^w $$ should be an isomorphism. Where by $ X^v $ we mean for example $ X^{(1,0,1)}=X_1 \otimes I \otimes X_3 $.

Note that here I make the somewhat unusual choice of excluding $ i $ from the qubit Pauli group (in other words generating it purely from $ X,Z $ type operators), cf. here

Answer 1

Yes, if $p$ is odd. This fact is well-known (at least among some people), and you can find it in some papers on the phase space formalism. Note that the Pauli group (also called Heisenberg-Weyl group for the obvious reason), is a unitary representation of the Heisenberg group of $\mathbb F_p^{2n}$ . This representation is sometimes called the Weyl representation, but I have also read Schrödinger representation (it's unitarily equivalent to the one you have written down). It's the central object to the discrete Stone-von Neumann theorem und the construction of the metaplectic/Weil representation of $\mathrm{Sp}_{2n}(\mathbb F_p)$.

However, if $p$ is even, the situation is (again) more complicated. First of all, the usual definition of the Heisenberg group does not extend to $\mathbb F_2$. The Pauli group is also not an extraspecial 2-group (it has order $2^{2n+2}$ and centre $\mathbb Z_4$). However, there is a way of defining a Heisenberg group as a non-trivial central extension of $\mathbb F_2^{2n}$ by $\mathbb Z_4$. It has a unitary representation which exactly gives the well-known multi-qubit Pauli group.

Fun fact: The real Pauli group (i.e. the Pauli matrices with real entries) is an extraspecial 2-group $2^{2n+1}$. That's what you get if you set $p=2$ in your construction (not including the $i$).

For details, feel free to have a look at my thesis (in particular Sec. 3.1, 3.2, and 4.1, as well as 6.2). I made some effort in collecting such statements ;)