  We study the relation between the Ising problem Hamiltonian parameters and
the minimum spectral gap (min-gap) of the system Hamiltonian in the Ising-based
quantum annealer. The main argument we use in this paper to assess the
performance of a QA algorithm is the presence or absence of an anti-crossing
during quantum evolution. For this purpose, we introduce a new parametrization
definition of the anti-crossing. Using the Maximum-weighted Independent Set
(MIS) problem in which there are flexible parameters (energy penalties J
between pairs of edges) in an Ising formulation as the model problem, we
construct examples to show that by changing the value of J, we can change the
quantum evolution from one that has an anti-crossing (that results in an
exponential small min-gap) to one that does not have, or the other way around,
and thus drastically change (increase or decrease) the min-gap. However, we
also show that by changing the value of $J$ alone, one can not avoid the
anti-crossing. We recall a polynomial reduction from an Ising problem to an MIS
problem to show that the flexibility of changing parameters without changing
the problem to be solved can be applied to any Ising problem. As an example, we
show that by such a reduction alone, it is possible to remove the anti-crossing
and thus increase the min-gap. Our anti-crossing definition is necessarily
scaling invariant as scaling the problem Hamiltonian does not change the nature
(i.e. presence or absence) of an anti-crossing. As a side note, we show exactly
how the min-gap is scaled if we scale the problem Hamiltonian by a constant
factor.
