Tensor networks can be thought of as linear maps between Hilbert spaces. They are used in physics to scale down huge calculations of many body systems, and have a strong mathematical basis. Tensor networks are basically multiple tensors connected together to form a network.
Take the tensor $ R_1 $ as a linear map between two Hilbert spaces $ H_t $ and $ H_s $, where $t_1, t_2, s_1, $ and $ s_2 $ are the labels of the indices of $ R_1 $:
$ R_1: H_{t_1} \otimes H_{t_2} \to H_{s_1} \otimes H_{s_2} $.
Another tensor $ R_2 $ is defined similarly as:
$ R_2: H_{t_3} \otimes H_{t_4} \to H_{s_3} \otimes H_{s_4} $.
$ R_1 $ and $ R_2 $ are rank four tensors because they have four indices.
$ R_1 $ and $ R_2 $ are "joined" to form a lattice of rank four tensors.
Say there are nine rank four tensors. So we can describe this in a set,
$ G_1 = (R_1,R_2,...,R_9), $
These nine rank four tensors form an inner cycle and each of them except the one at the center is joined to a tensor of rank five. The tensors of rank five are defined similarly to the tensors of rank four. Here is how $ E_1 $ is defined:
$ E_1: H_{t_5} \otimes H_{t_6} \to H_{s_5} \otimes H_{s_6} $.
And the set of these tensors is:
$ G_2 = (E_1,E_2,...,E_4), $.
The tensor $ E_1 $ is the rank five tensor a the top left corner of the network. The tensor $ E_2 $ is the rank five tensor in the lower right corner of the network.
Removing all interior tensors (the inner cycle of nine rank four tensors) we can view $ E_1 $ and $ E_2 $ as an inner product, correct? If we can, then I guess we can also view $ E_3 $ and $ E_4 $ alone as an inner product as well, where $ E_3 $ and $ E_4 $ are the upper right and lower left corners respectively.
Let the two inner products overlap orthogonally and treat each intersection as a rank four tensor. Is this network I described a valid tensor network?
What might be some strengths and weaknesses of this tensor network?
