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

Merging GW with DMFT and non-local correlations beyond

We review recent developments in electronic structure calculations that go beyond state-of-the-art methods such as density functional theory (DFT) and dynamical mean field theory (DMFT). Specifically, we discuss the following methods: GW as implemented in the Vienna {\it ab initio} simulation package (VASP) with the self energy on the imaginary frequency axis, GW+DMFT, and ab initio dynamical vertex approximation (D$\Gamma$A). The latter includes the physics of GW, DMFT and non-local correlations beyond, and allows for calculating (quantum) critical exponents. We present results obtained by the three methods with a focus on the benchmark material SrVO$_3$.Comment: tutorial review submitted to EPJ-ST (scientific report of research unit FOR 1346); 11 figures 27 page

Quantum Algorithms for Computational Geometry Problems

We study quantum algorithms for problems in computational geometry, such as POINT-ON-3-LINES problem. In this problem, we are given a set of lines and we are asked to find a point that lies on at least $3$ of these lines. POINT-ON-3-LINES and many other computational geometry problems are known to be 3SUM-HARD. That is, solving them classically requires time $\Omega(n^{2-o(1)})$, unless there is faster algorithm for the well known 3SUM problem (in which we are given a set $S$ of $n$ integers and have to determine if there are $a, b, c \in S$ such that $a + b + c = 0$). Quantumly, 3SUM can be solved in time $O(n \log n)$ using Grover's quantum search algorithm. This leads to a question: can we solve POINT-ON-3-LINES and other 3SUM-HARD problems in $O(n^c)$ time quantumly, for $c<2$? We answer this question affirmatively, by constructing a quantum algorithm that solves POINT-ON-3-LINES in time $O(n^{1 + o(1)})$. The algorithm combines recursive use of amplitude amplification with geometrical ideas. We show that the same ideas give $O(n^{1 + o(1)})$ time algorithm for many 3SUM-HARD geometrical problems.Comment: 10 page