  We study the Hamiltonian identifiability of a many-body spin-1/2 system
assisted by the measure- ment on a single quantum probe based on the
eigensystem realization algorithm (ERA) approach employed in Phys. Rev. Lett.
113, 080401 (2014). We demonstrate a potential application of Gr\"obner basis
to the identifiability test of the Hamiltonian, and provide the necessary
experimental resources, such as the lower bound in the number of the required
measurement points, the upper bound in total required evolution time, and thus
the total measurement time. Focusing on the examples of the identifiability in
the spin chain model with nearest-neighboring interaction, we classify the
spin-chain Hamiltonian based on its identifiability, and provide the control
protocols to engineer the nonidentifiable Hamiltonian to be an identifiable
Hamiltonian.

Quantum Physics