LE-QAOA: A Complexity Theory Enhanced Tool for Quantum Optimization

Combinatorial optimization problems are at the heart of modern computational challenges; however, their NP-hard nature often makes exact classical solutions intractable. The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising candidate for NISQ-era optimization. However, a significant performance gap exists between unconstrained and constrained formulations. The former refers to problems in which each solution is valid, either bad or good. The latter, to tasks in which there exist illegal solutions. While native constrained QAOA mixers offer high solution quality, they often suffer from prohibitive runtimes. In this paper, we introduce LE-QAOA (L-reduction Enhanced QAOA), a framework that leverages classical complexity theory to bridge this gap. By utilizing L-reductions to map diverse problems into a canonical Max2SAT form, we are able to solve constrained problems using efficient unconstrained subroutines followed by a classical repairing tournament. Our results on the Maximum Independent Set (MIS) problem show that LE-QAOA achieves solution quality that matches and frequently exceeds that of native constrained solvers, while requiring as little as 10% of the execution time. This approach effectively combines the scalability of unconstrained quantum evolution with the precision of constrained optimization. Moreover, this approach makes it possible to solve combinatorial optimization by reduction to a single chosen problem whose QAOA encoding is optimized, rather than explicitly building specific Hamiltonians.