What is the general matrix for the Swap gate?

Question

In section 3.3.2 of this PDF, The general SWAP gate is defined as

$
S (alpha, hat{y}) = begin{bmatrix} 
1  &  0  &  0  &  0  \
0  &  cos(alpha/2)  &  -sin(alpha/2)  &  0  \
0  &  sin(alpha/2) &  cos(alpha/2)  &  0  \
0  &  0  &  0  &  1 \
end{bmatrix}
$

The same lecture notes claim that for $alpha = pi$, you get the SWAP gate. This is not correct if we perform the computation.

$
S (pi, hat{y}) = begin{bmatrix} 
1  &  0  &  0  &  0  \
0  &  0  &  -1  &  0  \
0  &  1 &  0  &  0  \
0  &  0  &  0  &  1 \
end{bmatrix}
$

Those lecture notes also say the square root of SWAP can be created by setting $alpha=frac{pi}{2}$. When we do that we get

$
S (frac{pi}{2}, hat{y}) = begin{bmatrix} 
1  &  0  &  0  &  0  \
0  &  frac{1}{sqrt{2}}  &  -frac{1}{sqrt{2}}  &  0  \
0  &  frac{1}{sqrt{2}} &  frac{1}{sqrt{2}}  &  0  \
0  &  0  &  0  &  1 \
end{bmatrix}
$

The matrix for the square root of Swap is 
$
begin{bmatrix} 
1  &  0  &  0  &  0  \
0  &  frac{1}{{2}} (1+i)  &  frac{1}{{2}} (1-i)  &  0  \
0  &  frac{1}{{2}} (1-i)  &  frac{1}{{2}} (1+i)  &  0  \
0  &  0  &  0  &  1 \
end{bmatrix}
$

This is not the same matrix as the one we get when we use the general SWAP matrix. Is the matrix for the general SWAP from those lecture notes correct? I haven't been able to find another source to cross-reference.

Martin Vesely · Answer

A gate $S (alpha, hat{y})$ implements this circuit:

Here is an example of code for $alpha = pi/4$ (other parameters of $U3$ have to be set as stated):

cx q[1], q[0];
cu3(pi/4,-pi,pi) q[0],q[1];
cx q[1], q[0];

Setting $alpha = pi$ leads to something similar to swap gate up to a phase for input $|10rangle$ in which case $-|01rangle$ is returned.

What is the general matrix for the Swap gate?

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