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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?

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  • $\begingroup$ Welcome to Quantum Computing SE! I've made a minor edit to your question, so it now asks why it makes sense to develop them further, instead of asking if it does - people are researching them after all (although this is perhaps already contained in the second part of the question). Of course, feel free to re-edit if you don't like what I've changed $\endgroup$ – Mithrandir24601♦ Aug 29 '18 at 18:11
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    $\begingroup$ @Mithrandir24601 I think this is a major edit. Also, I think the question is too broad (it is really a list-like question, not to mention that "what are the current most pressing questions" will be outdated soon, at least we hope so). $\endgroup$ – Norbert Schuch Aug 29 '18 at 20:11
  • $\begingroup$ @Mithrandir24601 I mean, do you have any evidence that tensor networks are used to "pave the way towards quantum supremacy"? $\endgroup$ – Norbert Schuch Aug 29 '18 at 21:31
  • $\begingroup$ @NorbertSchuch It's entirely possible I'm reading the question wrong, but to me, that's half of what the question's asking with the other half asking about current research questions, which, while you could argue that it would make sense as a separate question, it's not like there are no other questions asking what current research is either - not quite the same as asking about the 'most pressing research questions', sure, but depending on how it's looked at, not a world apart either (although yes, it is on the broader side of questions) $\endgroup$ – Mithrandir24601♦ Aug 29 '18 at 22:05
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    $\begingroup$ I think this will really be helpful: arxiv.org/abs/1603.03039 $\endgroup$ – Siddhant Singh Nov 4 '18 at 16:33
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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.

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