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It is shown that the gap that limits the speed of a quantum annealing processcan take three salient morphologies: (i) the supersteep gap where both theground and the first excited eigenenergy level curves have topologicallyrelated pairs of nearby inflection points giving both the maximum and theminimum a steep aspect, (ii) the steep gap where only the first excitedeigenenergy level curve has a pair of inflection points giving its minimum asteep aspect while the maximum of the ground level does not exhibit inflectionpoints, and (iii) the mild gap that has no related inflection points.Classification of the various singularities betrayed by the inflection pointsrelies on the critical value curves of the quadratic numerical range mapping ofthe matrix H0+iH1, where H0 is the transverse field Hamiltonian and H1 theproblem Hamiltonian. It is shown that the ground level is mapped to thegenerically smooth boundary of the numerical range, while the first excitedlevel is mapped to an interior non-smooth critical value curve exhibitingswallow tails. The major result is that the position of the swallow tailsrelative to the boundary allows the supersteep versus steep discrimination,while the absence of swallow tail-boundary interaction characterizes the mildgap. As a corollary of the singularity analysis, the highly structured initialand final Hamiltonians of the Grover search create unstable singularities thatbreak into stable swallow tails under perturbation, with the consequence ofinvalidating the gap scaling estimates computed around the unstablesingularity. Classification of all stable singularities from a global viewpointrequires the Legendrian approach where the energy level curves becomeLegendrian knots in the contact space. Last but not least, it will be shownthat a supersteep swallow tail previews tunneling.

Topological Discrimination of Steep to Supersteep Gap to Emergence of Tunneling in Adiabatic Quantum Processes