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I have been active in this community a while and I repeatedly see questions and answers referring to the so-called Tensor Networks. That makes me wonder what are those elements are and what the motivation behind their construction is. References about where to look for the motivation and construction of those is sought in this question.

Additionally, an insight into the importance of these elements in the quantum computing and/or quantum communication paradigms would be helpful too, as the concept is found in this site in several places.

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    $\begingroup$ My feeling is that asking for a general introduction into a topic is beyond the scope of a Q&A site. You might try a review, e.g. Orus: arxiv.org/abs/1306.2164 $\endgroup$
    – Norbert Schuch
    Sep 3, 2018 at 23:18
  • $\begingroup$ Yeah, I know. Thanks for the reference, that's what I was looking for, references that help the reader to introduce to the concept. I will add the resource request tag again in order to clarify. $\endgroup$
    – Josu Etxezarreta Martinez
    Sep 4, 2018 at 7:15
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    $\begingroup$ another recent review is Biamonte and Bergholm: arxiv.org/abs/1708.00006 $\endgroup$
    – glS ♦
    Sep 6, 2018 at 16:38
  • $\begingroup$ Google now has TensorNetwork library opensourced: github.com/google/TensorNetwork The following paper is a good reference (also mentioned on github): arxiv.org/abs/1905.01330 $\endgroup$
    – Santanu Ganguly
    Aug 5, 2019 at 10:09
  • $\begingroup$ Here's another reference: arxiv.org/abs/1812.04011 $\endgroup$
    – mando
    Aug 6, 2019 at 13:20

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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 contraction allows you to pick which axes get paired up.

A tensor network is a configuration of tensors and contractions, drawn as a graph where the nodes are tensors and the contractions are edges.

Quantum circuits are a kind of tensor network. The gates are matrices (a specific kind of tensor) and the qubits are series of contractions (lines connecting the tensors). In some cases even the vertical lines in a circuit diagram (e.g. the line from a CNOT's control to its target) correspond to some sort of contraction, though circuit notation has strayed far enough from tensor network notation that this is not always the case.

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