Hamiltonian Learning is a process of recovering system Hamiltonian from measurements, which is a fundamental problem in quantum information processing. In this study, we investigate the problem of learning the symmetric Hamiltonian from its eigenstate. Inspired by the application of group theory in block diagonal secular determination, we have derived a method to determine the number of linearly independent equations about the Hamiltonian unknowns obtained from an eigenstate. This number corresponds to the degeneracy of the associated irreducible representation of the Hamiltonian symmetry group. To illustrate our approach, we examine the XXX Hamiltonian and the XXZ Hamiltonian. We first determine the Hamiltonian symmetry group, then work out the decomposition of irreducible representation, which serves as foundation for analyzing the uniqueness of recovered Hamiltonian. Our numerical findings consistently align with our theoretical analysis.

Hamiltonian Learning (HL) is essential for validating quantum systems in quantum computing. Not all Hamiltonians can be uniquely recovered from a steady state. HL success depends on the Hamiltonian model and the steady state. Here, we analyze HL for a specific type of steady state composed of eigenstates with degenerate mixing weight, making these Hamiltonian's eigenstates indistinguishable. To overcome this challenge, we utilize the orthogonality relationship between the eigenstate space and its complement space, constructing the orthogonal space equation. By counting the number of linearly independent equations derived from a steady state, we determine the recoverability of a generic local Hamiltonian. Our scheme is applicable for generic local Hamiltonians under various steady state, therefore offering a way of measuring the degree to which a steady state characterizes a Hamiltonian.