  Adiabatic quantum computing and optimization have garnered much attention
recently as possible models for achieving a quantum advantage over classical
approaches to optimization and other special purpose computations. Both
techniques are probabilistic in nature and the minimum gap between the ground
state and first excited state of the system during evolution is a major factor
in determining the success probability. In this work we investigate a strategy
for increasing the minimum gap and success probability by introducing
intermediate Hamiltonians that modify the evolution path between initial and
final Hamiltonians. We focus on an optimization problem relevant to recent
hardware implementations and present numerical evidence for the existence of a
purely local intermediate Hamiltonian that achieve the optimum performance in
terms of pushing the minimum gap to one of the end points of the evolution. As
a part of this study we develop a convex optimization formulation of the search
for optimal adiabatic schedules that makes this computation more tractable, and
which may be of independent interest. We further study the effectiveness of
random intermediate Hamiltonians on the minimum gap and success probability,
and empirically find that random Hamiltonians have a significant probability of
increasing the success probability, but only by a modest amount.
