Classification of two-qudit Clifford hierarchy gates

Let $d$ be the qudit dimension, and consider the Clifford hierarchy gates of two qudits. Let $H_i$ denote the Hadamard gate on the $i$th qudit, let $E_i$ and $S_i$ denote the qudit and qubit phase gates, respectively, and let $CZ$ denote the controlled-Z gate. Let $D$ and $C$ be the gates appearing in the one-qudit semi-Clifford normal form. Define

$$D_1=(E_1)^a,$$

and

$$D_2=(CZ)^b(E_2)^c,$$

for $a,b,c d$. In the qubit case, replace $E_1,E_2$ by $S_1,S_2$ and take $a,b,c d$.

Classification of two-qudit Clifford hierarchy gates. Every hierarchy gate of two qudits is either a Clifford gate or can be uniquely expressed as

$$M_1M_2DC,$$

where $M_1$ is $H_1$ or $H_1D_1H_1^*$, and $M_2$ is $H_2$ or $H_1H_2D_2H_2^*H_1^*$.

The statement is presented as a proposed direction for future work extending the one-qudit classification, so its resolution is not established in the supplied text.