  Determining the low-energy eigenspectra of quantum many-body systems is a
long-standing challenge in physics. In this work, we solve this problem by
introducing two novel algorithms to determine low-energy eigenstates based on a
compact matrix product state (MPS) representation of the multiple targeted
eigenstates. The first algorithm utilizes a canonicalization approach that
takes advantage of the imaginary-time evolution of multi-target MPS, offering
faster convergence and ease of implementation. The second algorithm employs a
variational approach that optimizes local tensors on the Grassmann manifold,
capable of achieving higher accuracy. These algorithms can be used
independently or combined to enhance convergence speed and accuracy. We apply
them to the transverse-field Ising model and demonstrate that the calculated
low-energy eigenspectra agree remarkably well with the exact solution.
Moreover, the eigenenergies exhibit uniform convergence in gapped phases,
suggesting that the low-energy excited eigenstates have nearly the same level
of accuracy as the ground state. Our results highlight the accuracy and
versatility of multi-target MPS-based algorithms for determining low-energy
eigenspectra and their potential applications in quantum many-body physics.
