Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$ we canonically associate permanent of $M$ to the number of perfect matchings in the graph.

Is there a bipartite graph which is balanced we can associate to the square of the determinant or absolute value of the determinant in a canonical manner without computing the determinant explicitly?

Update Observe the construction of the graph is non-canonical in any known way. It has to be clever.