  The standard approach to encoding constraints in quantum optimization is the
quadratic penalty method. Quadratic penalties introduce additional couplings
and energy scales, which can be detrimental to the performance of a quantum
optimizer. In quantum annealing experiments performed on a D-Wave Advantage, we
explore an alternative penalty method that only involves linear Ising terms and
apply it to a customer data science problem. Our findings support our
hypothesis that the linear Ising penalty method should improve the performance
of quantum optimization compared to using the quadratic penalty method due to
its more efficient use of physical resources. Although the linear Ising penalty
method is not guaranteed to exactly implement the desired constraint in all
cases, it is able to do so for the majority of problem instances we consider.
For problems with many constraints, where making all penalties linear is
unlikely to be feasible, we investigate strategies for combining linear Ising
penalties with quadratic penalties to satisfy constraints for which the linear
method is not well-suited. We find that this strategy is most effective when
the penalties that contribute most to limiting the dynamic range are removed.
