Black-box optimization (BBO) can be used to optimize functions whose analytic form is unknown. A common approach to realising BBO is to learn a surrogate model which approximates the target black-box function which can then be solved via white-box optimization methods. In this paper, we present our approach BOX-QUBO, where the surrogate model is a QUBO matrix. However, unlike in previous state-of-the-art approaches, this matrix is not trained entirely by regression, but mostly by classification between “good” and “bad” solutions. This better accounts for the low capacity of the QUBO matrix, resulting in significantly better solutions overall. We tested our approach against the state-of-the-art on four domains and in all of them BOX-QUBO showed better results. A second contribution of this paper is the idea to also solve white-box problems, i.e. problems which could be directly formulated as QUBO, by means of black-box optimization in order to reduce the size of the QUBOs to the information-theoretic minimum. Experiments show that this significantly improves the results for MAX-k-SAT.

The Stable Marriage Problem (SMP) describes the problem, of finding a stable matching between two equally sized sets of elements (e.g., males and females) given an ordering of preferences for each element. A matching is stable, when there does not exist any match of a male and female which both prefer each other to their current partner under the matching. Finding such a matching of maximum cardinality, when ties and incomplete preference lists are allowed, is called MAX-SMTI and is an NP-hard variation of the SMP. In this work a Quadratic Unconstrained Binary Optimization (QUBO) formulation for MAX-SMTI is introduced and solved both with D-Wave Systems quantum annealing hardware and by their classical meta-heuristic QBSolv. Both approaches are reviewed against existing state-of-the-art approximation algorithms for MAX-SMTI. Additionally, the proposed QUBO problem can also be used to count stable matchings in SMP instances, which is proven to be a #P-complete problem. The results show, that the proposed (quantum) methods can compete with the classical ones regarding the solution quality and might be a relevant alternative, when quantum hardware scales with respect to the number of qubits and their connectivity.