One can receive some lower bounds on $N(d)$ from the theory of groups of central type. Let then $G$ be a finite group of order $d^2$. Assume that there exists a non-degenerate two cocycle $[\alpha]\in H^2(G,\mathbb{C}^{\times})$. This means that the twisted group algebra $\mathbb{C}^{\alpha}G$ is isomorphic with $M_d(\mathbb{C})$. This also means that $M_d(\mathbb{C})$ will have a basis $\{U_g\}_{g\in G}$ such that $U_gU_h=\alpha(g,h)U_{gh}$. Take now $X$ to be $\{U_{g_1},\ldots U_{g_r}\}$ where $\{g_1,\ldots ,g_r\}$ is a generating set for $G$. Then $\mathcal{A}_n(X)=M_d(\mathbb{C})$, and $N(d)$ will be the length of a maximal path in the (oriented) Cayley graph of $G$ with respect to this generating set (where we have an oriented edge from $g$ to $gs$, for $s$ in the generating set).

From this point on we do not really to take $\alpha$ into account, we just need to know that it exists. The group $\mathbb{Z}/d\times\mathbb{Z}/d$ is a group of central type. With respect to the standard basis $\{(0,1),(1,0)\}$ we have a Cayley graph with longest path of length $2d-2$. That shows already that $N(d)$ is at least $2d-2$.

I do not know if the question about the length of paths in Cayley graphs of finite groups of central type was studied. I believe, however, that you can get better lower bounds for $N(d)$ from this.

One can receive some lower bounds on $N(d)$ from the theory of groups of central type. Let then $G$ be a finite group of order $d^2$. Assume that there exists a non-degenerate two cocycle $[\alpha]\in H^2(G,\mathbb{C}^{\times})$. This means that the twisted group algebra $\mathbb{C}^{\alpha}G$ is isomorphic with $M_d(\mathbb{C})$. This also means that $M_d(\mathbb{C})$ will have a basis $\{U_g\}_{g\in G}$ such that $U_gU_h=\alpha(g,h)U_{gh}$. Take now $X$ to be $\{U_{g_1},\ldots U_{g_r}\}$ where $\{g_1,\ldots ,g_r\}$ is a generating set for $G$. Then $\mathcal{A}_n(X)=M_d(\mathbb{C})$, and $N(d)$ will be the maximal length of a geodesic in the (oriented) Cayley graph of $G$ with respect to this generating set (where we have an oriented edge from $g$ to $gs$, for $s$ in the generating set).

From this point on we do not really to take $\alpha$ into account, we just need to know that it exists. The group $\mathbb{Z}/d\times\mathbb{Z}/d$ is a group of central type. With respect to the standard basis $\{(0,1),(1,0)\}$ we have a Cayley graph with maximal length of geodesic $2d-2$. That shows already that $N(d)$ is at least $2d-2$.

I do not know if the question about the length of geodesics in Cayley graphs of finite groups of central type was studied. I believe, however, that you can get better lower bounds for $N(d)$ from this.