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

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.Comment: 32 pages. Minor revisions and corrections. Regev and Schiff's proof that P(OR) = \Omega(N) remove

Microseismicity Simulated on Asperity-Like Fault Patches: On Scaling of Seismic Moment With Duration and Seismological Estimates of Stress Drops

Observations show that microseismic events from the same location can have similar source durations but different seismic moments, violating the commonly assumed scaling. We use numerical simulations of earthquake sequences to demonstrate that strength variations over seismogenic patches provide an explanation of such behavior, with the event duration controlled by the patch size and event magnitude determined by how much of the patch area is ruptured. We find that stress drops estimated by typical seismological analyses for the simulated sources significantly increase with the event magnitude, ranging from 0.006 to 8 MPa. However, the actual stress drops determined from the on‐fault stress changes are magnitude‐independent and ~3 MPa. Our findings suggest that fault heterogeneity results in local deviations in the moment‐duration scaling and earthquake sources with complex shapes of the ruptured area, for some of which stress drops may be significantly (~100–1,000 times) underestimated by the typical seismological methods