Upper Bounds on Quantum Query Complexity Inspired by the Elitzur-Vaidman Bomb Tester

Inspired by the Elitzur-Vaidman bomb testing problem [Elitzur/Vaidman 1993], we introduce a new query complexity model, which we call bomb query complexity B(f). We investigate its relationship with the usual quantum query complexity Q(f), and show that B(f)=Theta(Q(f)^2). This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on Q(f)=Theta(sqrt(B(f))). We subsequently were able to give explicit quantum algorithms matching our upper bound method. We apply this method on the single-source shortest paths problem on unweighted graphs, obtaining an algorithm with O(n^(1.5)) quantum query complexity, improving the best known algorithm of O(n^(1.5) * sqrt(log(n))) [Furrow, 2008]. Applying this method to the maximum bipartite matching problem gives an O(n^(1.75)) algorithm, improving the best known trivial O(n^2) upper bound