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To date, research in quantum computation promises potential for outperformingclassical heuristics in combinatorial optimization. However, when aiming atprovable optimality, one has to rely on classical exact methods like integerprogramming. State-of-the-art integer programming algorithms can compute strongrelaxation bounds even for hard instances, but may have to enumerate a largenumber of subproblems for determining an optimum solution. If the potential ofquantum computing realizes, it can be expected that in particular findinghigh-quality solutions for hard problems can be done fast. Still, near-futurequantum hardware considerably limits the size of treatable problems. In thiswork, we go one step into integrating the potentials of quantum and classicaltechniques for combinatorial optimization. We propose a hybrid heuristic forthe weighted maximum-cut problem or, equivalently, for quadratic unconstrainedbinary optimization. The heuristic employs a linear programming relaxation,rendering it well-suited for integration into exact branch-and-cut algorithms.For large instances, we reduce the problem size according to a linearrelaxation such that the reduced problem can be handled by quantum machines oflimited size. Moreover, we improve the applicability of QAOA, a parameterizedquantum algorithm, by deriving optimal parameters for special instances whichmotivates a parameter estimate for arbitrary instances. We present numerouscomputational results from real quantum hardware.

Enhancing Quantum Algorithms for Quadratic Unconstrained Binary Optimization via Integer Programming