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Communications in Theoretical Physics

A Parallel Quantum Algorithm for the Satisfiability Problem

Liu Wen-Zhang, Zhang Jing-Fu, Long Gui-Lu

Published Mar 1, 2008• 5 citations
Publisher

Abstract

In this paper we present a classical parallel quantum algorithm for the satisfiability problem. We have exploited the classical parallelism of quantum algorithms developed in [G.L. Long and L. Xiao, Phys. Rev. A 69 (2004) 052303], so that additional acceleration can be gained by using classical parallelism. The quantum algorithm first estimates the number of solutions using the quantum counting algorithm, and then by using the quantum searching algorithm, the explicit solutions are found.

DOI
10.1088/0253-6102/49/3/22

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A previously developed quantum search algorithm for solving 1-SAT problems in a single step is generalized to apply to a range of highly constrained k-SAT problems. We identify a bound on the number of clauses in satisfiability problems for which the generalized algorithm can find a solution in a constant number of steps as the number of variables increases. This performance contrasts with the linear growth in the number of steps required by the best classical algorithms, and the exponential number required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems in fact have no solutions, unlike previously proposed quantum search algorithms.
Boolean satisfiability (SAT) solving is a fundamental problem in computer science. Finding efficient algorithms for SAT solving has broad implications in many areas of computer science and beyond. Quantum SAT solvers have been proposed in the literature based on Grover's algorithm. Although existing quantum SAT solvers can consider all possible inputs at once, they evaluate each clause in the formula one by one sequentially, making the time complexity O(m) -- linear to the number of clauses m -- per Grover iteration. In this work, we develop a parallel quantum SAT solver, which reduces the time complexity in each iteration from linear time O(m) to constant time O(1) by utilising extra entangled qubits. To further improve the scalability of our solution in case of extremely large problems, we develop a distributed version of the proposed parallel SAT solver based on quantum teleportation such that the total qubits required are shared and distributed among a set of quantum computers (nodes), and the quantum SAT solving is accomplished collaboratively by all the nodes. We have proved the correctness of our approaches and demonstrated them in simulations.