  Stabilizer states have been commonly utilized in quantum information, quantum
error correction, and quantum circuit simulation due to their simple
mathematical structure. In this work, we apply stabilizer states to tackle
quantum many-body problems and introduce the concept of stabilizer ground
states. We establish an equivalence formalism for identifying stabilizer ground
states of general Pauli Hamiltonians. Moreover, we also develop an exact and
linear-scaled algorithm to obtain stabilizer ground states of 1D local
Hamiltonians and thus free from discrete optimization. This proposed
equivalence formalism and linear-scaled algorithm are not only applicable to
finite-size systems, but also adaptable to infinite periodic systems. The
scalability and efficiency of the algorithms are numerically benchmarked on
different Hamiltonians. Finally, we demonstrate that stabilizer ground states
are promising tools for not only qualitative understanding of quantum systems,
but also cornerstones of more advanced quantum state ansatzes on both classical
and quantum computers.
