A tensor network is a representation of a tensor by contracting selected indices
of smaller tensors according to a graph.
Each internal edge labels a summed index,
while uncontracted indices label the output tensor.
For example, the matrix product 
contracts the index 
of two tensors.
The dimension of an internal index
is its bond dimension.
A projected entangled-pair state uses one tensor at each site of a lattice, with internal indices on lattice edges and one output index for the local Hilbert space. A small bond dimension limits the number of parameters but does not itself guarantee an accurate approximation to a particular ground state.
For a finite region of the square grid with fixed boundary colors, a network of four-index tensors can count tilings by Wang tiles. Each cell tensor records the allowed color tuples, and tensor contraction sums over the colors of shared sides (Liang 2026).
