Recover vector $x$ from rank-$1$ matrix $Q=xx^H$ Let the matrix $Q \in\mathbb{C}^{n \times n}$ be known. It is also known that $Q=xx^H$, where $x=[x_1,\ldots,x_n]^T$ and $x^H$ is its conjugate transpose. What is $x$? How to recover it?

If $Q=xx^H$, then $Qx=xx^Hx$. Set $\lambda=x^Hx$; this means $x$ is an eigenvector of $Q$ relative to $\lambda$ (the unique non zero eigenvalue).
On the other hand, if $Q$ is explicitly given, then since it has rank one, you just take a nonzero column $y$ and consider that $x=\mu y$ for some $\mu$. Then do $Q=xx^H=|\mu|^2yy^H$ and you can determine $\mu$ (up to a complex number of modulo $1$). (Assuming $x\ne0$, of course, or the problem is trivial).
