I am trying to understand what the importance of tensor networks is (or will/could be) for quantum computing.
Does it make sense to study tensor networks deeply and develop them further to help pave the way towards quantum supremacy? If so, how could they help and what are the current most pressing research questions?
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.
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.
