  We utilize the theory of local amplitude transfers (LAT) to gain insights
into quantum walks (QWs) and quantum annealing (QA) beyond the adiabatic
theorem. By representing the eigenspace of the problem Hamiltonian as a
hypercube graph, we demonstrate that probability amplitude traverses the search
space through a series of local Rabi oscillations. We argue that the amplitude
movement can be systematically guided towards the ground state using a
time-dependent hopping rate based solely on the problem's energy spectrum.
Building upon these insights, we extend the concept of multi-stage QW by
introducing the guided quantum walk (GQW) as a bridge between QW-like and
QA-like procedures. We assess the performance of the GQW on exact cover,
traveling salesperson and garden optimization problems with 9 to 30 qubits. Our
results provide evidence for the existence of optimal annealing schedules,
beyond the requirement of adiabatic time evolutions. These schedules might be
capable of solving large-scale combinatorial optimization problems within
evolution times that scale linearly in the problem size.
