Algorithms for variational Monte Carlo calculations of fermion PEPS in the swap gates formulation

In recent years, the variational Monte Carlo (VMC) calculations of projected entangled pair states (PEPS) has emerged as a competitive method for computing the ground states of many-body quantum systems. This method is particularly important for fermion systems where sign problems are abundant. We derive and explain the algorithms for the VMC calculations of fermion PEPS in the swap gates formulation. It is the purpose of this paper to be as concise and precise as possible. Diagramatic tensor notation is used whenever possible to serve this purpose. As a separate result, we prove the detailed balance of sequential sampling of tensor networks.