Are there formulas for sums of products of unitary matrices of the form $\sum_i U_{ij}U_{il}U_{ik}^* U_{im}^*$?

Question

If $U$ is unitary, then by definition $$\sum_i U_{ij}U_{ik}^* = \delta_{jk}\tag1$$ holds.

What about products of more than two copies of the same unitary matrix? Consider for example the following product of four copies of the same unitary matrix:

$$\sum_i U_{ij}U_{il}U_{ik}^* U_{im}^*.\tag2$$ Can something be said in this case? Does a variation of (1) hold?