  With the development of quantum many-body simulator, Hamiltonian tomography
has become an increasingly important technique for verification of quantum
devices. Here we investigate recovering the Hamiltonians of two spin chains
with 2-local interactions and 3-local interactions by measuring local
observables. For these two models, we show that when the chain length reaches a
certain critical number, we can recover the local Hamiltonian from its one
steady state by solving the homogeneous operator equation (HOE) developed in
Ref. [1]. To explain the existence of such a critical chain length, we develop
an alternative method to recover Hamiltonian by solving the energy eigenvalue
equations (EEE). By using the EEE method, we completely recovered the numerical
results from the HOE method. Then we theoretically prove the equivalence
between the HOE method and the EEE method. In particular, we obtain the
analytical expression of the rank of the constraint matrix in the HOE method by
using the EEE method, which can be used to determine the correct critical chain
length in all the cases.
