solQHealer: Quantum Procedures for Rendering Infeasible Solutions Feasible: A Proof of Concept with the Maximum Independent Set Problem and 3-SAT

Over the past decade, the usefulness of quantum annealing hardware for combinatorial optimization has been the subject of active debate. Although current analog quantum machines do not guarantee optimality, operating instead as heuristic solvers, the technology is evolving rapidly. Beyond performance alone, this emerging technologies offers fundamentally new approaches to problem-solving that are not readily accessible to classical exact methods particularly in dynamic environments or online optimization settings. This paper focuses on one such approaches: Reverse Quantum Annealing (RQA). Unlike classical exact methods, RQA allows the optimization process to begin from an initial infeasible solution by embedding it directly into the qubits’ initial state. We leverage this capability by formulating problem constraints as penalty terms within Quadratic Unconstrained Binary Optimization (QUBO) models, thereby preserving infeasible solutions within the quantum search space. We propose iterative strategies that apply RQA in three distinct modes to rapidly repair infeasible solutions. These methods are evaluated on two well-known NP-hard problems: the Maximum Independent Set (MIS) and the 3-SAT problem. Our results demonstrate the effectiveness of RQA in steering infeasible configurations toward feasibility, offering promising potential for real-time applications where problem data may change suddenly and solutions must be repaired swiftly.