How to construct an $ntimes n$ unitary matrix taking an arbitrary $|psirangle$ to a target state $|phirangle$?

I didn't go through the attached pdf. But if you want to find a unitary matrix $U$ that maps a quantum state $|psi rangle$ to $|phirangle$ then you can use the Householder transformation as I commented. Here the two vectors have the same length (they are unit vectors) because we are thinking of them as a quantum state, so there will always exist a Householder transformation that can do this.

For example: If you want to find a unitary, $U$, that maps $|00rangle = begin{pmatrix} 1 0 0 0 end{pmatrix}$ to $|11rangle = begin{pmatrix} 01end{pmatrix}$ then you can construct it as: $$ U = I - 2vv^T $$ where $v$ is the normalized vector of $|00rangle - |11rangle = begin{pmatrix} 1 0 0 -1 end{pmatrix}$. That is, $ v = begin{pmatrix} 1/sqrt{2} 0 0 -1/sqrt{2} end{pmatrix} $.

From here, we can write $U$ out explicitly as:

begin{align} U &= begin{pmatrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1end{pmatrix} - 2 begin{pmatrix} 1/sqrt{2} 0 0 -1/sqrt{2} end{pmatrix} begin{pmatrix} 1/sqrt{2} & 0 & 0 &-1/sqrt{2} end{pmatrix} &= begin{pmatrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1end{pmatrix} - 2 begin{pmatrix} 1/2 & 0 & 0 & -1/2 0 & 0 & 0 & 0 0 & 0 & 0 & 0 -1/2 & 0 & 0 & 1/2 end{pmatrix} = begin{pmatrix} 0 & 0 & 0 & 1 0 & 1 & 0 & 0 0 & 0 & 1 & 0 1 & 0 & 0 & 0end{pmatrix} end{align}

You can check that this is infact unitary since $Ucdot U^dagger = I$ and that

$$ U|00rangle = begin{pmatrix} 0 & 0 & 0 & 1 0 & 1 & 0 & 0 0 & 0 & 1 & 0 1 & 0 & 0 & 0end{pmatrix} begin{pmatrix} 1 0 0 0 end{pmatrix} = begin{pmatrix} 0 0 0 1 end{pmatrix} = |11rangle $$

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The question now is about how to decompose this unitary matrix into a quantum circuit with certain set of gates... This can be done in different ways... look up KAK decomposition if you are interested.