Some finite quantum Riemannian geometries

I outline a formulation of quantum Riemannian geometry using bimodule connections. I give some examples in the finite case where it is possible for a fixed differential algebra to construct the moduli of all possible quantum Riemannian geometries on it, as a step towards quantum gravity. In particular, this will be solved for the geometry of a square graph, including a reasonable Einstein-Hilbert action for the quantum metric as given by lengths assigned to the edges.