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

Abstract: We present an exact quantum algorithm for solving the Exact Satisfiability (XSAT) problem, which belongs to the important NP-complete complexity class. The algorithm is based on an intuitive approach that can be divided into two parts: First, the identification and efficient characterization of a restricted subspace that contains all the valid assignments of the XSAT; Second, a quantum search in such restricted subspace. The quantum algorithm can be used either to find a valid assignment (or to certify that no solution exists) or to count the total number of valid assignments. The query complexities for the worst-case are respectively bounded by $O(\sqrt{2^{n-M^{\prime}}})$ and $O(2^{n-M^{\prime}})$, where $n$ is the number of variables and $M^{\prime}$ 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 XSAT. As a concrete application, we provide the worst-case complexity for the Hamiltonian cycle problem obtained after mapping it to a suitable XSAT. Specifically, we show that the time complexity for the proposed quantum algorithm is bounded by $O(2^{n/4})$ for 3-regular undirected graphs, where $n$ is the number of nodes. The same worst-case complexity holds for $(3,3)$-regular bipartite graphs (the current best classical algorithm has a (worst-case) running time bounded by $O(2^{31n/96})$). Finally, when compared to heuristic techniques for XSAT, the proposed quantum algorithm is faster than the classical WalkSAT 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 proposed quantum algorithm can be also extended to the generalized version of the XSAT known as Occupation problem.