  It is shown that the gap that limits the speed of a quantum annealing process
can take three salient morphologies: (i) the supersteep gap where both the
ground and the first excited eigenenergy level curves have topologically
related pairs of nearby inflection points giving both the maximum and the
minimum a steep aspect, (ii) the steep gap where only the first excited
eigenenergy level curve has a pair of inflection points giving its minimum a
steep aspect while the maximum of the ground level does not exhibit inflection
points, and (iii) the mild gap that has no related inflection points.
Classification of the various singularities betrayed by the inflection points
relies on the critical value curves of the quadratic numerical range mapping of
the matrix H0+iH1, where H0 is the transverse field Hamiltonian and H1 the
problem Hamiltonian. It is shown that the ground level is mapped to the
generically smooth boundary of the numerical range, while the first excited
level is mapped to an interior non-smooth critical value curve exhibiting
swallow tails. The major result is that the position of the swallow tails
relative to the boundary allows the supersteep versus steep discrimination,
while the absence of swallow tail-boundary interaction characterizes the mild
gap. As a corollary of the singularity analysis, the highly structured initial
and final Hamiltonians of the Grover search create unstable singularities that
break into stable swallow tails under perturbation, with the consequence of
invalidating the gap scaling estimates computed around the unstable
singularity. Classification of all stable singularities from a global viewpoint
requires the Legendrian approach where the energy level curves become
Legendrian knots in the contact space. Last but not least, it will be shown
that a supersteep swallow tail previews tunneling.
