Abstract
We study the costs and benefits of different quantum approaches to finding approximate solutions of constrained combinatorial optimization problems with a focus on the maximum independent set. Using the Lagrange multiplier approach, we analyze the dependence of the output on graph density and circuit depth. The Quantum Alternating Operator Ansatz approach is then analyzed, and we examine the dependence on different choices of initial states. This approach, although powerful, is expensive in terms of quantum resources. We also introduce a new algorithm, the dynamic quantum variational ansatz (DQVA), that dynamically adapts 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.
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