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So far I have read a little bit about zx-calculus & y-calculus.

From Reversible Computation:

The zx-calculus is a graphical language for describing quantum systems.

The zx-calculus is an equational theory, based on rewriting the diagrams which comprise its syntax. Re-writing can be automated by means of the quantomatic software.

This method seems very interesting, however I am not able to find much introductory information on the subject. Any insight into the subject or additional resources would be greatly appreciated.

Current Resources:

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    $\begingroup$ By introductory information, do you mean like about string diagrams for monoidal categories in general? $\endgroup$ – AHusain Jul 16 '18 at 0:48
  • $\begingroup$ @AHusain Upon further investigation, it seems so (added link) $\endgroup$ – user820789 Jul 16 '18 at 1:15
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The best possible textbook reference at the moment is

It is written by one of the two inventors of the ZX calculus (Bob Coecke), and one of the people who has contributed the most to the development of Quantomatic (Aleks Kissinger), and so would be the definitive introductory reference.

There is now also a website, zxcalculus.com, with tutorials, links to resources, and exhibiting a Python-based tool called PyZX, which you may find helpful. This website is a coordinated effort by the main proponents and developers of the theory and applications of the ZX calculus, of which I am one.

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You already put Selinger's survey, so here are a couple more links.

Baez and Stay: Baez and Stay is a survey article. It covers monoidal, braided, symmetric and dagger categories. For the example related to quantum computation focus on either Hilb or cobordism. The appropriate string diagrams for these are included along with the sections for those types of categories. It points the connections between logic and type theory as well, but you don't need those sections. However, it would be helpful. You could see Baez's other blog posts as well.

Qiaochu Yuan's blog: Qiaochu's blog post is more introductory and brief. It focuses solely on the vector space example as to avoid prerequisites besides linear algebra. The later posts in that series cover other adjectives to add on such as braided, symmetric or dagger. See later in that series as well.

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