Let $P_n$ be the Pauli group on n qubits. The order of the group is $|P_n| = 2^{2n+2}$. I beleive the outer automorphism group is closely related to the symplectic group $Sp(2n,2)$ but I don't know the exact relation. Direct calculations for small n seem to show that $|Out(P_n)| = 2 |Sp(2n,2)|$. Does anyone know of how this can shown in general or a reference on the subject.
2026-03-30 08:15:01.1774858501
What is the outer automorphism group of the Pauli group?
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It looks like some work was done a while ago by Michael Planat on the outer automorphism groups of the Pauli group [1]. He has some other resources that may be good to look at as well [2], and you could also look on Google Scholar for other papers / collaborators to start a more thorough search.
Sources
Planat, Michel, and Maurice Kibler. "Unitary reflection groups for quantum fault tolerance." Journal of computational and theoretical nanoscience 7.9 (2010): 1759-1770. Link here
“Reflection Groups for Quantum Computing” slide deck, Michael Planat. Link here