This chapter introduces the tensor network ansatz for a quantum state whose entanglement entropy obeys the so-called area law with or without logarithmic corrections. This ansatz represents a quantum many-body wave function by a network product of local tensors defined on the lattice sites and treats all tensor elements as variational parameters. It includes, for example, one-dimensional matrix product states (MPS) and two-dimensional projected entangled pair states (PEPS) or projected entangled simplex states (PESS). A typical example is the spin-1 AKLT chain, whose ground state can be exactly represented as an MPS. If a logarithmic correction to the entanglement area law emerges, a tensor network state termed the multi-scale entanglement renormalization ansatz (MERA) describes the entanglement structure of the ground state more accurately in one dimension.

This chapter discusses the methods of solving PEPS or other two-dimensional tensor network states, including variational optimization and the annealing simulation. The variation optimization determines the local terms by minimizing the ground-state energy. The annealing simulation takes the full or simple update strategy to filter out the ground state through the imaginary time evolution. The nonlinear effect arises in evaluating the derivative of uniform PEPS and is avoided by utilizing automatic differentiation. Both variational optimization and the annealing simulation involve a contraction of double-layer tensor network states. This contraction is the primary technical barrier in the study of PEPS. A nested tensor network approach is introduced to combat this difficulty.