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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?

Mauricio
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  • 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 – Mark Spinelli Sep 29 '22 at 22:18
  • @MarkS I meant a group as in group theory, what is the group there? isnt that clifford+non-cliffod=universal – Mauricio Sep 29 '22 at 22:54
  • See here https://quantumcomputing.stackexchange.com/questions/2036/why-are-quantum-gates-unitary-and-not-special-unitary – Mark Spinelli Sep 30 '22 at 02:29
  • @MarkS are you pointing to (special) unitary groups? That is not discrete. – Mauricio Sep 30 '22 at 07:41
  • Would you consider a discrete universal gateset trivial? If not you have ofc e.g. ${H, T, CNOT}$ – JSdJ Sep 30 '22 at 11:42
  • @Mauricio sorry I missed that part of the question – Mark Spinelli Sep 30 '22 at 11:48
  • If you consider the Cliffords to be finite, (basically by removing its center $U(1)$), then I think every set of the Clifford Hierarchy is also finite. – JSdJ Sep 30 '22 at 11:50
  • @JSdJ good point. I guess I am looking for sets that do not form a universal gate set. Also I am not considering necessarily finite sets. – Mauricio Sep 30 '22 at 16:10
  • Every finite group occurs as a subgroup of a unitary group. Any faithful representation of it gives an embedding. I think you mean to add more restrictions (contain non trivial subgroup of Paulis?....) – unknown Sep 30 '22 at 19:08
  • @unknown not necessarily finite groups, also must include a non-Clifford gate so it cannot be a subgroup of Pauli – Mauricio Sep 30 '22 at 19:24
  • @Mauricio I said *contain" a subgroup of Paulis... – unknown Sep 30 '22 at 19:59

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