Faster than Classical Quantum Algorithm for dense Formulas of Exact Satisfiability and Occupation Problems

We present an exact quantum algorithm for solving the Exact Satisfiability problem, which is known to belong to the important NP-complete complexity class. The algorithm is based on an intuitive approach that can be divided into two parts: The first step consists in the identification and efficient characterization of a restricted subspace that contains all the valid assignments (if any solution exists) of the Exact Satisfiability; while the second part performs a quantum search in such restricted subspace. The quantum algorithm can be used either to find a valid assignment (or to certify that no solutions exist) or to count the total number of valid assignments. The query complexity for the worst-case is, respectively, bounded by O( √ 2n−M) and O(2n−M ′ ), where n is the number of variables and M ′ the number of linearly independent clauses. Remarkably, the proposed quantum algorithm results to be faster than any known exact classical algorithm to solve dense formulas of Exact Satisfiability. When compared to heuristic techniques, the proposed quantum algorithm is faster than the classical WalkSAT heuristic and Adiabatic Quantum Optimization for random instances with a density of constraints close to the satisfiability threshold, the regime in which instances are typically the hardest to solve. The quantum algorithm that we propose can be straightforwardly extended to the generalized version of the Exact Satisfiability known as Occupation problems. The general version of the algorithm is presented and analyzed.