  The identifiability of a system is concerned with whether the unknown
parameters in the system can be uniquely determined with all the possible data
generated by a certain experimental setting. A test of quantum Hamiltonian
identifiability is an important tool to save time and cost when exploring the
identification capability of quantum probes and experimentally implementing
quantum identification schemes. In this paper, we generalize the
identifiability test based on the Similarity Transformation Approach (STA) in
classical control theory and extend it to the domain of quantum Hamiltonian
identification. We employ STA to prove the identifiability of spin-1/2 chain
systems with arbitrary dimension assisted by single-qubit probes. We further
extend the traditional STA method by proposing a Structure Preserving
Transformation (SPT) method for non-minimal systems. We use the SPT method to
introduce an indicator for the existence of economic quantum Hamiltonian
identification algorithms, whose computational complexity directly depends on
the number of unknown parameters (which could be much smaller than the system
dimension). Finally, we give an example of such an economic Hamiltonian
identification algorithm and perform simulations to demonstrate its
effectiveness.
