Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002],
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)=Θ(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)=Θ(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(n1.5) quantum
query complexity, improving the best known algorithm of O(n1.5logn) [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite
matching problem gives an O(n1.75) algorithm, improving the best known
trivial O(n2) upper bound.Comment: 32 pages. Minor revisions and corrections. Regev and Schiff's proof
that P(OR) = \Omega(N) remove