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 all of the qubits in mind, in a tensor product vector space of all the state spaces of the individual qubits.
A quantum circuit can be understood as a product of linear operators on some quantum state and here lies the power of tensor networks. These diagrams provide a visual way of understanding these products which also allows for visual manipulation in a rigorous way allowing one to desgin algorithms in a novel way or look at information problems in a new light.
A Visual Way to Understand Linear Algebra Problems
This is the power of tensor networks.
I invite you to check out this document which really goes through these ideas I am motivating https://arxiv.org/pdf/1603.03039.pdf, particularly how quantum teleportation can be thought of as a tensor network.