Quantum Algorithm for Dynamic Programming Approach for DAGs and Applications

In this paper, we present a quantum algorithm for the dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is O(√(n̂m)logn̂) , and the running time of the best known deterministic algorithm is O(n+m) , where n is the number of vertices, n̂ is the number of vertices with at least one outgoing edge; m is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX, and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluation. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.