Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester

TL;DR

This paper introduces a new quantum query complexity model inspired by the Elitzur-Vaidman bomb tester, establishing a quadratic relationship with standard quantum complexity and applying it to improve algorithms for shortest paths and bipartite matching.

Contribution

The paper defines bomb query complexity, proves its quadratic relation to quantum query complexity, and uses this to develop improved quantum algorithms for specific graph problems.

Findings

Established $B(f)= heta(Q(f)^2)$ relationship

Developed a method to derive quantum algorithms from classical ones

Improved quantum query bounds for shortest paths and bipartite matching

Abstract

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 $B(f)=\Theta(Q(f)^2)$. 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 $O(n^{1.5}\sqrt{\log n})$ [arXiv:quant-ph/0606127]. 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.