  We investigate the irreconcilability issue that raises in translating the
search algorithm from the Continuous-Time Quantum Walk (CTQW) framework to the
Adiabatic Quantum Computing (AQC) framework. For the AQC formulation to evolve
along the same path as the CTQW requires a constant energy gap in the former
Hamiltonian throughout the AQC schedule. To resolve the issue, we modify the
CTQW-inspired AQC catalyst Hamiltonian with a $Z$ oracle operator. Through
simulation we demonstrate that the total running time for the proposed approach
remains optimal. Inspired by this solution, we further investigate adaptive
scheduling for the catalyst Hamiltonian and its coefficient function in the
adiabatic path to improve the adiabatic local search.
