Explore an in-depth analysis of matchgate tensor networks in this comprehensive lecture from the Thematic Programme on "Tensor Networks: Mathematical Structures and Novel Algorithms." Delve into the Hamiltonian description of 2D matchgates tensor networks with open physical boundaries, and uncover the analytical understanding of a previously numerically observed phase transition. Examine the fascinating correspondence between a many-body topological phase transition in a Z_2 gauge theory (toric code) and a topological single-particle transition in a class D superconductor. Further investigate random matchgate tensor networks and learn about the conditions under which an ensemble of tensor networks can be described by a non-linear super matrix model. Gain valuable insights into advanced concepts in quantum physics and tensor network theory throughout this 75-minute presentation.