I will discuss recent exact solutions for non-equilibrium steady state density operators of several boundary driven quantum chains, namely XXZ spin 1/2, Fermi-Hubbard, and the Lai-Sutherland spin-1 chains, with the aim of establishing a unifying framework. The infinite bond-dimension matrix product operator for the steady states in all cases can be neatly encoded in terms of operator sums of walks over particular infinite graphs. In some cases, (local) Lax structure can be identified, corresponding to the integrability of the problem.