How to construct an $n\times n$ unitary matrix taking an arbitrary $|\psi\rangle$ to a target state $|\phi\rangle$?

Question

I came across Lecture 12 here https://viterbi-web.usc.edu/~tbrun/Course/ that does this but I was not able to understand. An example would be very helpful

Answer 1

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 $|\phi\rangle$ 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 $|00\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}$ to $|11\rangle = \begin{pmatrix} 0\\0\\0\\1\end{pmatrix}$ then you can construct it as: $$ U = I - 2vv^T $$ where $v$ is the normalized vector of $|00\rangle - |11\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}$. 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 & 1\end{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 & 1\end{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 & 0\end{pmatrix} \end{align}

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

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

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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.