  We study the unlabeled sensing problem that aims to solve a linear system of
equations $A x =\pi(y) $ for an unknown permutation $\pi$. For a generic matrix
$A$ and a generic vector $y$, we construct a system of polynomial equations
whose unique solution satisfies $ A\xi^*=\pi(y)$. In particular, $\xi^*$ can be
recovered by solving the rank-one moment matrix completion problem. We propose
symbolic and numeric algorithms to compute the unique solution. Some numerical
experiments are conducted to show the efficiency and robustness of the proposed
algorithms.
