Smallest Circuit for Square of Sparse Symmetric Matrix
If it's really depth that you care about, I think you should be able to do it with a circuit of depth $O(\log^2 n)$ using sorting networks.  I'll assume we want to compute the product $AB$, and that $A$ is represented as an array of rows, where each row is a sorted list of non-zero entries; and $B$ is an array of columns, where each column is a sorted list of non-zero entries.  Then the $ik$ cell of the product $AB$ (namely, $(AB)_{ik}$) can be computed as follows: copy the entire row $A_{i\cdot}$; copy the entire column $B_{\cdot k}$; concatenate these two lists; sort them based on their indices, using a sorting network, to bring together entries with a common index; at each position where both matrices have an entry, multiply those entries; and sum up those products.  You can build a sorting network with depth $O(\log^2 n)$, and the multiply-and-sum step can be done with depth $O(\log n)$ using an addition tree.