Quantum Algorithm for Shortest Path Search in Directed Acyclic Graph

Abstract

In this work, we consider a well-known “Single Source Shortest Path Search” problems for weighted directed acyclic graphs (DAGs). We suggest a quantum algorithm with time complexity \(O(\sqrt {nm} \,\log \;n)\) and O(1/n) error probability, where n is a number of Vertexes, m is the number of edges. We use quantum dynamic programming approach (Khadiev, 2018) and Dürr and Høyer minimum search algorithm to speed up our search. Our algorithm is better than C. Dürr and coauthors’ quantum algorithm for an undirected graph. The time complexity of C. Dürr’s algorithm is \(O(\sqrt {nm} \,{(\log \;n)^2})\). The best known deterministic algorithm for the problem is based on a dynamic programming approach and its time complexity is O (n + m). At the same time, if we use algorithms for general graphs, then we do not get better results. The time complexity of best implementations of Dijkstra’s algorithm with Fibonacci heap is O (m + n log n).