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

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

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 $$i$$).
• 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! May 13 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. May 16 at 8:04
• 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 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 $p \neq 2$? May 17 at 20:09
• @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. May 18 at 7:00