Quick question about Two-qubit SWAP gate from the Exchange interaction

Question

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:

DaftWullie · Accepted Answer

You need to calculate $U=e^{-iHt}$. The trick to doing this is working out the eigenvectors of $H$: there's $|00rangle$ and $|11rangle$ with eigenvalues J, and
$$
|Psi^{pm}rangle=(|01ranglepm|10rangle)/sqrt{2}
$$
with eigenvalues $(-1pm 2)J$. In particular, notice that this means 3 of the eigenvalues are $J$. Hence, there are two eigenspaces of $H$, $|Psi^-ranglelanglePsi^-|$ and $I-|Psi^-ranglelanglePsi^-|$. Hence, we can find
$$
U=e^{-iJt}(I-|Psi^-ranglelanglePsi^-|)+e^{3iJt}|Psi^-ranglelanglePsi^-|.
$$
If you remove an irrelevant global phase, this is just the same as
$$
U=(I-|Psi^-ranglelanglePsi^-|)+e^{4iJt}|Psi^-ranglelanglePsi^-|.
$$
This is exactly what you were after, with $4Jt=pialpha$.

Quick question about Two-qubit SWAP gate from the Exchange interaction

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