  Determining the energy gap in a quantum many-body system is critical to
understanding its behavior and is important in quantum chemistry and condensed
matter physics. The challenge of determining the energy gap requires
identifying both the excited and ground states of a system. In this work, we
consider preparing the prior distribution and circuits for the eigenspectrum of
time-independent Hamiltonians, which can benefit both classical and quantum
algorithms for solving eigenvalue problems. The proposed algorithm unfolds in
three strategic steps: Hamiltonian transformation, parameter representation,
and classical clustering. These steps are underpinned by two key insights: the
use of quantum circuits to approximate the ground state of transformed
Hamiltonians and the analysis of parameter representation to distinguish
between eigenvectors. The algorithm is showcased through applications to the 1D
Heisenberg system and the LiH molecular system, highlighting its potential for
both near-term quantum devices and fault-tolerant quantum devices. The paper
also explores the scalability of the method and its performance across various
settings, setting the stage for more resource-efficient quantum computations
that are both accurate and fast. The findings presented here mark a new insight
into hybrid algorithms, offering a pathway to overcoming current computational
challenges.
