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If I have a quantum gate, how can I get the gate that represents half-way of that gate? As an example of what I want: How do I derive $\sqrt {SWAP} $ gate

$$\sqrt {SWAP} = \left( {\matrix{ 1 & 0 & 0 & 0 \cr 0 & {{{1 + i} \over 2}} & {{{1 - i} \over 2}} & 0 \cr 0 & {{{1 - i} \over 2}} & {{{1 + i} \over 2}} & 0 \cr 0 & 0 & 0 & 1 \cr } } \right)$$ from $SWAP$ gate? $$SWAP = \left( {\matrix{ 1 & 0 & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 0 & 1 \cr } } \right)$$

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    $\begingroup$ by "half-way" gate do you just mean square root? $\endgroup$
    – Condo
    Oct 27 '21 at 15:26
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    $\begingroup$ Many versions of this question have been asked. (I asked one myself recently!) But I like in particular this one. $\endgroup$
    – Mark S
    Oct 27 '21 at 16:20
  • $\begingroup$ Thank you all for your attention. The answer that Mark S attached is exactly what I was looking for. $\endgroup$
    – Bekaso
    Oct 27 '21 at 16:32

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