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I do not know many groups (as in group theory) of quantum gates. Aside from trivial ones, I know there is Pauli group and the Clifford group. Recently I discovered another interesting group generated by $\sqrt{X}$ and controlled-$\sqrt{X}$ (NCV library). There is also the NOT, CNOT, Toffoli (NCT library).

What examples of groups that are

  • discrete,
  • have two or more generators,
  • are not the same as the groups listed above but with a different Pauli gate,
  • have at least a non-Clifford gate, and
  • are not considered a universal gate set (like $\{H,T,\mathrm{CNOT}\}$)

are there?

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  • $\begingroup$ Google's Sycamore uses the roots of Pauli gates (along with $W=(X+Y)/\sqrt 2$) as single-qubit gates, and a modified $\mathsf{iSWAP}$ as a two-qubit gate. See here and also here $\endgroup$ Sep 29, 2022 at 22:18
  • $\begingroup$ @MarkS I meant a group as in group theory, what is the group there? isnt that clifford+non-cliffod=universal $\endgroup$
    – Mauricio
    Sep 29, 2022 at 22:54
  • $\begingroup$ See here quantumcomputing.stackexchange.com/questions/2036/… $\endgroup$ Sep 30, 2022 at 2:29
  • $\begingroup$ @MarkS are you pointing to (special) unitary groups? That is not discrete. $\endgroup$
    – Mauricio
    Sep 30, 2022 at 7:41
  • $\begingroup$ Would you consider a discrete universal gateset trivial? If not you have ofc e.g. $\{H, T, CNOT\}$ $\endgroup$
    – JSdJ
    Sep 30, 2022 at 11:42

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