Efficient Digital Quadratic Unconstrained Binary Optimization Solvers for SAT Problems

Boolean satisfiability (SAT) is a propositional logic problem of determining whether an assignment of variables satisfies a Boolean formula. Many combinatorial optimization problems can be formulated in Boolean SAT logic – either as k-SAT decision problems or Max k-SAT optimization problems, with conflict-driven (CDCL) solvers being the most prominent. Despite their ability to handle large instances, CDCL-based solvers have fundamental scalability limitations. In light of this, we propose recently-developed quadratic unconstrained binary optimization (QUBO) solvers as an alternative platform for 3-SAT problems. To utilize them, we implement a 2-step [3-SAT]-[Max 2-SAT]-[QUBO] conversion procedure and present a rigorous proof to explicitly calculate the number of both satisfied and violated clauses of the original 3-SAT instance from the transformed Max 2-SAT formulation. We then demonstrate, through numerical simulations on several benchmark instances, that digital QUBO solvers can achieve state-of-the-art accuracy on 78-variable 3-SAT benchmark problems. Our work facilitates the broader use of quantum annealers on noisy intermediate-scale quantum (NISQ) devices, as well as their quantum-inspired digital counterparts, for solving 3-SAT problems.

💡 Research Summary

The paper tackles the longstanding challenge of scaling Boolean satisfiability (SAT) solving beyond the limits of modern conflict‑driven clause learning (CDCL) solvers by leveraging recently developed quadratic unconstrained binary optimization (QUBO) techniques. The authors propose a concrete two‑step reduction pipeline: first, each 3‑SAT clause is transformed into a set of ten 2‑SAT clauses using the (7, 10)‑gadget, introducing a single auxiliary variable per original clause; second, the resulting Max‑2‑SAT instance is encoded as a QUBO problem. The key insight of the gadget is that a satisfied 3‑SAT clause yields exactly seven satisfied clauses among the ten derived 2‑SAT clauses, while an unsatisfied clause yields six. This deterministic relationship enables a rigorous back‑mapping from the solution of the QUBO model to the original 3‑SAT instance.

In the Max‑2‑SAT‑to‑QUBO conversion, each clause is represented by signed coefficients vₖⱼ∈{−1,0,1} that capture literal polarity. By constructing a quadratic penalty term that counts violated clauses, the authors derive an explicit Q matrix and linear vector b for the QUBO objective q(x)=½ xᵀQx + bᵀx + c. The formulation is exact (up to a constant shift) and can be fed to any digital QUBO heuristic.

A central theoretical contribution is the proof that, once the QUBO solver returns an optimal binary vector, the numbers of satisfied and violated original 3‑SAT clauses can be recovered by solving a simple linear Diophantine equation: S′ = 7·S + 6·V, where S′ is the total number of satisfied 2‑SAT clauses, and S and V are the unknown satisfied/violated counts for the original problem. The authors extend this reasoning to cases where auxiliary variables take mixed values (0 or 1), showing that the original clause statistics can still be reconstructed by partitioning the transformed instance and applying the same linear relationship to each part.

Empirically, the authors benchmark several state‑of‑the‑art digital QUBO solvers (including Ising‑based approaches) against RC2, a leading CDCL‑based Max‑SAT solver. Four publicly available 3‑SAT benchmarks (78 variables, clause density 1.6–6) and a suite of randomly generated instances spanning clause densities 0.5–6, including the notoriously hard region, are used. The “best” QUBO heuristic is identified as a genetic algorithm with local search, selected by median violated‑clause count. Across all test sets, the QUBO approach matches or slightly exceeds RC2’s accuracy; for the 78‑variable benchmark the QUBO solver achieves 99.2 % clause satisfaction, essentially indistinguishable from RC2.

The work demonstrates that QUBO solvers—both quantum‑inspired digital and, by extension, quantum annealers—can serve as viable alternatives to CDCL for solving 3‑SAT, provided a careful reduction is applied. By establishing a provable, lossless mapping between SAT and QUBO, the authors enable exact recovery of SAT solutions from QUBO outputs, a property essential for practical deployment. The paper opens several avenues for future research: integrating the pipeline with near‑term quantum annealers, optimizing the gadget to reduce overhead (currently a ten‑fold clause blow‑up), and scaling the approach to larger SAT instances where CDCL struggles with memory and parallelism. Overall, the study provides both theoretical foundations and experimental evidence that QUBO‑based optimization can compete with the current SAT‑solving paradigm.