If I understand what you are asking for (I don't know much about tensor networks), both equations are singular value decompositions of $U$, just with respect to different indices (and in the latter case highlighting the singular values, which in the first equation is "hidden" in $A,B$).
There are two things to notice here. First of all, given an arbitrary ...
A Different Way Of Looking At Linear Algebra
Tensor Networks provide a different way of looking at linear algebra particularly within the context of tensor space systems.
Quantum Circuits Are Just Products of Vectors and Operators
A quantum circuit is inherently a tensor space system as when we have multiple qubits we must think of the whole circuit with ...
A tensor is a multi-dimensional array. A vector is a 1d tensor, a matrix is a 2d tensor, and so forth.
Tensor contraction is a generalization of matrix multiplication. In matrix multiplication, you pair up the second axis of the first matrix with the first axis of the second matrix and sum over their pointwise product to produce new entries. A tensor ...
By treating a quantum circuit as a network model, the network model can be optimized the order of the calculation by making a contract between tensor and tensor.
I think it is definitely important to learn TN for future qc.