  We describe a general methodology for enhancing the efficiency of adiabatic
quantum computations (AQC). It consists of homotopically deforming the original
"Hamiltonian surface" in a way that the redistribution of the Gaussian
curvature weakens the effect of the anti-crossing, thus yielding the desired
improvement. Our approach is not pertubative but instead is built on our
previous global description of AQC in the language of Morse theory. Through the
homotopy deformation we witness the birth and death of critical points whilst,
in parallel, the Gauss-Bonnet theorem reshuffles the curvature around the
changing set of critical points. Therefore, by creating enough critical points
around the anti-crossing, the total curvature--which was initially centered at
the original anti-crossing--gets redistributed around the new neighbouring
critical points, which weakens its severity and so improves the speedup of the
AQC. We illustrate this on two examples taken from the literature.
