  We study the costs and benefits of different quantum approaches to finding
approximate solutions of constrained combinatorial optimization problems with a
focus on Maximum Independent Set. In the Lagrange multiplier approach we
analyze the dependence of the output on graph density and circuit depth. The
Quantum Alternating Ansatz Approach is then analyzed and we examine the
dependence on different choices of initial states. The Quantum Alternating
Ansatz Approach, although powerful, is expensive in terms of quantum resources.
A new algorithm based on a "Dynamic Quantum Variational Ansatz" (DQVA) is
proposed that dynamically changes to ensure the maximum utilization of a fixed
allocation of quantum resources. Our analysis and the new proposed algorithm
can also be generalized to other related constrained combinatorial optimization
problems.

Quantum Physics