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:

Answer 1

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