Solution
= Solution
The three <controlled-NOT gate>[controlled-NOT gates] have directions $1\to2$, $2\to1$, and $1\to2$. On <computational basis> bits they act as
$$
(x,y)\mapsto(x,x\oplus y)
\mapsto(y,x\oplus y)
\mapsto(y,x).
$$
By linearity, the whole <quantum circuit> is the <SWAP gate>. Therefore
$$
\boxed{U(|\psi\rangle\otimes|\phi\rangle)
=|\phi\rangle\otimes|\psi\rangle},
$$
so $|\alpha\rangle=|\phi\rangle$ and $|\beta\rangle=|\psi\rangle$.