  Hamiltonian simulation is a domain where quantum computers have the potential
to outperform their classical counterparts due to their inherent quantum
behavior. One of the main challenges of such quantum algorithms is up-scaling
the system size, which is necessary to achieve meaningful quantum advantage. In
this work, we present an approach to improve the scalability of eigenspace
filtering for the ground state preparation of a given Hamiltonian. Our method
aims to tackle limitations introduced by a small spectral gap and high
degeneracy of low energy states. It is based on an adaptive sequence of
eigenspace filtering through Quantum Eigenvalue Transformation of Unitary
Matrices (QETU) followed by spectrum profiling. By combining our proposed
algorithm with state-of-the-art phase estimation methods, we achieved good
approximations for the ground state energy with local, two-qubit gate
depolarizing probability up to $10^{-4}$. To demonstrate the key results in
this work, we ran simulations with the transverse-field Ising Model on
classical computers using Qiskit. We compare the performance of our approach
with the static implementation of QETU and show that we can consistently
achieve three to four orders of magnitude improvement in the absolute error
rate.
