Cliffordness of the qutrit Hadamard gate

Consider a simple generalization of the Hadamard gate to qutrits, defined as follows.

$$$$\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} &0 \\ 0 &0&1 \end{pmatrix}$$$$

It is well known that the two qubit Hadamard gate is a Clifford gate. Is the above gate also a Clifford gate? I could not verify if so.

• So do you consider a general Pauli group here or something? Since, normally, isn't Clifford group define to be the normalizer of the Pauli group? which is in dim of power of 2. Aug 24 '21 at 20:09
• The Pauli group can be defined for any dimension; same for the Clifford group. Since the OP mentions "qutrit" in the title (not qubit) I think the setting is meant to be dim=3. Aug 25 '21 at 0:48
• Yes, the setting is dimension 3. Aug 25 '21 at 10:34
• why do you call this a generalization of the Hadamard gate? It doesn't really have many (if any) properties in common with it, aside from it having the form $H\oplus (1)$
– glS
Aug 25 '21 at 12:06
• It behaves like the Hadamard gate for $|0\rangle$ and $|1\rangle$. The terminology is borrowed from this paper: arxiv.org/pdf/1105.5485.pdf (Section II). Aug 25 '21 at 12:08

X=[[0,1,0],[0,0,1],[1,0,0]]