What is the set of generators for the qutrit Clifford group?

Question

According to this article, any Clifford gate, acting on $n$ qubits, can be generated by Hadamard, CNOT, and S gates.

What are the set of generators for qutrit Cliffords?

Answer 1

From the paper Normal form for single-qutrit Clifford+T operators and synthesis of single-qutrit gates, the Clifford group in $p>2$ dimensions acting on a sigle qudit is generated by $S$ and $H$ given by:

$$ \begin{gather} S=\sum_{j=0}^{p-1}\omega^{j(j+1)2^{-1}}|j\rangle\langle j| \\ H = \frac{1}{\sqrt{p}}\sum_{j=0}^{p-1}\sum_{k=0}^{p-1}\omega^{jk}|j\rangle\langle k| \end{gather} $$

where $j$ and $k$ denote elements of the finite field $\mathbb{Z}_p$ with multiplication and addition defined modulo $p$, $2^{-1}$ is an element of the same field, and $\omega$ denotes a $p$th rooth of unity $\omega^p=1$. You can set $p=3$ for your case.

Now, for the equivalent of the $CNOT$ gate you want to look into this answer.