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For a d dimensional single qudit, knowing the generators of the group being the Hadamard gate and the Phase gate, how would I generate the entire group computationally in python? Or for any finite group with 2 generators and their given matrices?

For example in the qutrit case (since the group will be relatively small compared to higher dimensional qudits), the Hadamard gate can be defined as $ H_3 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 1 & 1 \\ 1 & \omega & \omega^2 \\ 1 & \omega^2 & \omega^4 \end{pmatrix}$ and the phase gate is defined as $S_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \omega^{-2} & 0 \\ 0 & 0 & 1 \end{pmatrix}$ where $\omega = e^{i\frac{2\pi}{3}}$, $H_3^{4} = I$ and $S_3^{3}= I$, following from this paper https://arxiv.org/pdf/1603.02286. I understand the cardinality of this group should be 216, ignoring the global phases, but I am not quite sure what the best way is to construct the elements of that group computationally.

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  • $\begingroup$ The generators for the clifford group should also include the qudit pauli Z. If you can enumerate the generators and their relations then something like the Todd Coxeter algorithm might be helpful. $\endgroup$
    – ChrisD
    Commented May 14 at 23:59

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