Quantum Approximation II. Sobolev Embeddings

A basic problem of approximation theory, the approximation of functions from the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered from the point of view of quantum computation. We determine the quantum query complexity of this problem (up to logarithmic factors). It turns out that in certain regions of the domain of parameters p,q,r,d quantum computation can reach a speedup of roughly squaring the rate of convergence of classical deterministic or randomized approximation methods. There are other regions were the best possible rates coincide for all three settings.Comment: 23 pages, paper submitted to the Journal of Complexit

On a Problem in Quantum Summation

We consider the computation of the mean of sequences in the quantum model of computation. We determine the query complexity in the case of sequences which satisfy a $p$-summability condition for $1\le p<2$. This settles a problem left open in Heinrich (2001).Comment: 21 pages, paper submitted to the Journal of Complexit

Quantum Integration in Sobolev Classes

We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes $W^r_p([0,1]^d)$ and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Novak on integration of functions from H\"older classes.Comment: Paper submitted to the Journal of Complexity. 28 page

Controlling the 2p2p Hole Alignment in Neon via the 2s2s-3p3p Fano Resonance

We study the state-resolved production of neon ion after resonant photoionization of Ne via the $2s$-$3p$ Fano resonance. We find that by tuning the photon energy across the Fano resonance a surprisingly high control over the alignment of the final $2p$ hole along the polarization direction can be achieved. In this way hole alignments can be created that are otherwise very hard to achieve. The mechanism responsible for this hole alignment is the destructive interference of the direct and indirect (via the autoionizing $2s^{-1}3p$ state) ionization pathways of $2p$. By changing the photon energy the strength of the interference varies and $2p$-hole alignments with ratios up to 19:1 between $2p_0$ and $2p_{\pm 1}$ holes can be created: an effect normally only encountered in tunnel ionization using strong-field IR pulses. Including spin-orbit interaction does not change the qualitative feature and leads only to a reduction in the alignment by $2/3$. Our study is based on a time-dependent configuration-interaction singles (TDCIS) approach which solves the multichannel time-dependent Schr\"odinger equation.Comment: 7 pages, 4 figure

Randomized Complexity of Parametric Integration and the Role of Adaption I. Finite Dimensional Case

We study the randomized $n$-th minimal errors (and hence the complexity) of vector valued mean computation, which is the discrete version of parametric integration. The results of the present paper form the basis for the complexity analysis of parametric integration in Sobolev spaces, which will be presented in Part 2. Altogether this extends previous results of Heinrich and Sindambiwe (J.\ Complexity, 15 (1999), 317--341) and Wiegand (Shaker Verlag, 2006). Moreover, a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting is solved.Comment: 30 page