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I am reading the following paper: Optimal two-qubit quantum circuits using exchange interactions.

I have a problem with the calculation of the unitary evolution operator $U$ (Maybe it is stupid):

I have figure out the matrix of $H$:

\begin{equation} H = J \begin{bmatrix}1 & 0 & 0 & 0\\ 0 & -1 & 2 & 0\\ 0 & 2 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \end{equation}

But I cannot write the matrix of Operator $U$ and get the result of $(SWAP)^α$.

Could you please help me to calculate it? I really want to know how to get the matrix of U.

Thank you so much.

The figure is shown as below:

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Generally speaking, it is frowned upon to include pictures with text in questions, as it becomes hard to copy etc. Also, if imgur fails, the question becomes more or less unreadable. Furthermore, check https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference for a quick tutorial on how to perform proper markup for math.

– JSdJ – 2020-11-17T13:20:34.593Thank you, it is my first time to use this forum. I will, thanks again! – None – 2020-11-17T14:09:53.110