  Local Hamiltonians arise naturally in physical systems. Despite its seemingly
`simple' local structure, exotic features such as nonlocal correlations and
topological orders exhibit in eigenstates of these systems. Previous studies
for recovering local Hamiltonians from measurements on an eigenstate
$|\psi\rangle$ require information of nonlocal correlation functions. In this
work, we develop an algorithm to determine local Hamiltonians from only local
measurements on $|\psi\rangle$, by reformulating the task as an unconstrained
optimization problem of certain target function of Hamiltonian parameters, with
only polynomial number of parameters in terms of system size. We also develop a
machine learning-based-method to solve the first-order gradient used in the
algorithm. Our method is tested numerically for randomly generated local
Hamiltonians and returns promising reconstruction in the desired accuracy. Our
result shed light on the fundamental question on how a single eigenstate can
encode the full system Hamiltonian, indicating a somewhat surprising answer
that only local measurements are enough without additional assumptions, for
generic cases.
