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The Quantum Mathematica add-on package is a very good tool to implement quantum gates. It contains gates as shown in this figure.

enter image description here

How to use these gates to decompose square root SWAP gate?

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    $\begingroup$ are you asking for a gate decomposition of the square root swap in terms of the gates listed in the figure, or are you asking how to write said gate in Mathematica in general? In the latter case, can't you simply do swap=SparseArray@Thread[{{1,1}, {2,3}, {3,2}, {4,4}} -> 1]; MatrixFunction[Sqrt, swap]? $\endgroup$
    – glS
    Oct 28, 2021 at 8:10
  • $\begingroup$ @glS yes I want to decompose square root swap gate using the gates listed in the figure $\endgroup$
    – Bekaso
    Oct 28, 2021 at 11:00
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    $\begingroup$ so the question doesn't really have much to do with Mathematica.. $\endgroup$
    – glS
    Oct 28, 2021 at 12:53

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Why not use $$ e^{i\pi/4}(I+i SWAP)/\sqrt{2}? $$

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