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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). See Definition of the Pauli group and the Clifford group

Ian Gershon Teixeira
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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 ;)

Markus Heinrich
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  • Great answer as usual! I would love to know more about the real Pauli group. Is the corresponding Clifford group the group $ \mathcal{C}m=2^{1+2m}+.O^+(2m, 2) $ from Nebe Raines and Sloane? I suppose $ \mathcal{C}m $ is an orthogonal 2 design on $ O{2^n}(\mathbb{R}) $ but not a unitary 2 design on $ U_{2^n} $? Anyway I should probably just ask a separate question... Also thanks for recommending your thesis! I was reading part one this morning its really lovely! – Ian Gershon Teixeira May 13 '22 at 15:39
  • @IanGershonTeixeira thanks for the feedback :) Yes, it is the one from Nebe, Rains, and Sloane, cp. p. 64 in my thesis. BTW you can also construct a $O^-$ version. About the design question, the real Clifford group should not be a unitary 2-design (at the very minimum, $O^+(2n,2)$ has to act transitively on $\mathbb F_2^{2n}\setminus 0$, which it does not). Perhaps it is an orthogonal design, but I am not familiar with the rep theory in this case. – Markus Heinrich May 16 '22 at 08:04
  • You are right of course that the standard Heisenberg group defined via a symplectic form $ [;,;] $ with multiplication $ (v,t)(w,s)=(v+w,s+t+\frac{1}{2}[v,w]) $ is not well defined in characteristic $ 2 $. But what about the group of upper triangular matrices called $ H_{2n+1} $ and given in the "higher dimensions" section here https://en.wikipedia.org/wiki/Heisenberg_group. This is well defined in characteristic $ 2 $. And it seems to me that $ H_{2n+1}(\mathbb{F}_p) $ should coincide with the real Pauli Group for $ p=2 $ and with the standard Pauli group for all $ p \neq 2 $? – Ian Gershon Teixeira May 17 '22 at 20:09
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    @IanGershonTeixeira That's exactly a central extension given by the cocycle $\eta(v,w) = v_z \cdot w_x$ w.r.t. to some fixed polarization $\mathbb F_p^{2n} = L_z \oplus L_x$. For odd $p$, the resulting group is isomorphic to one defined by the cocycle $2^{-1} [v,w]$, in particular, the according representations result both in the standard Pauli group. For even $p$, however, the automorphisms are restricted to $O^+(2n,2)$ since $\eta(v,v)=Q^+(v)$. And yes, the according representation is the real Pauli group. See also the discussion around Eq. (6.23) in my thesis. – Markus Heinrich May 18 '22 at 07:00