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Given a Boolean formula $\phi(x)$ in conjunctive normal form (CNF), thedensity of states counts the number of variable assignments that violateexactly $e$ clauses, for all values of $e$. Thus, the density of states is ahistogram of the number of unsatisfied clauses over all possible assignments.This computation generalizes both maximum-satisfiability (MAX-SAT) and modelcounting problems and not only provides insight into the entire solution space,but also yields a measure for the \emph{hardness} of the problem instance.Consequently, in real-world scenarios, this problem is typically infeasibleeven when using state-of-the-art algorithms. While finding an exact answer tothis problem is a computationally intensive task, we propose a novel approachfor estimating density of states based on the concentration of measureinequalities. The methodology results in a quadratic unconstrained binaryoptimization (QUBO), which is particularly amenable to quantum annealing-basedsolutions. We present the overall approach and compare results from the D-Wavequantum annealer against the best-known classical algorithms such as theHamze-de Freitas-Selby (HFS) algorithm and satisfiability modulo theory (SMT)solvers.

Estimating the Density of States of Boolean Satisfiability Problems on Classical and Quantum Computing Platforms