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Quantum algorithms have been widely studied in the context of combinatorialoptimization problems. While this endeavor can often analytically andpractically achieve quadratic speedups, theoretical and numeric studies remainlimited, especially compared to the study of classical algorithms. We proposeand study a new class of hybrid approaches to quantum optimization, termedIterative Quantum Algorithms, which in particular generalizes the RecursiveQuantum Approximate Optimization Algorithm. This paradigm can incorporate hardproblem constraints, which we demonstrate by considering the MaximumIndependent Set (MIS) problem. We show that, for QAOA with depth $p=1$, thisalgorithm performs exactly the same operations and selections as the classicalgreedy algorithm for MIS. We then turn to deeper $p>1$ circuits and other waysto modify the quantum algorithm that can no longer be easily mimicked byclassical algorithms, and empirically confirm improved performance. Our workdemonstrates the practical importance of incorporating proven classicaltechniques into more effective hybrid quantum-classical algorithms.

Iterative Quantum Algorithms for Maximum Independent Set: A Tale of Low-Depth Quantum Algorithms