  We introduce two methods for speeding up adiabatic quantum computations by
increasing the energy between the ground and first excited states. Our methods
are even more general. They can be used to shift a Hamiltonian's density of
states away from the ground state, so that fewer states occupy the low-lying
energies near the minimum, hence allowing for faster adiabatic passages to find
the ground state with less risk of getting caught in an undesired low-lying
excited state during the passage. Even more generally, our methods can be used
to transform a discrete optimization problem into a new one whose unique
minimum still encodes the desired answer, but with the objective function's
values forming a different landscape. Aspects of the landscape such as the
objective function's range, or the values of certain coefficients, or how many
different inputs lead to a given output value, can be decreased *or* increased.
One of the many examples for which these methods are useful is in finding the
ground state of a Hamiltonian using NMR: If it is difficult to find a molecule
such that the distances between the spins match the interactions in the
Hamiltonian, the interactions in the Hamiltonian can be changed without at all
changing the ground state. We apply our methods to an AQC algorithm for integer
factorization, and the first method reduces the maximum runtime in our example
by up to 754%, and the second method reduces the maximum runtime of another
example by up to 250%. These two methods may also be combined.
