Quantum Fourier transform revisited

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view

Simple processors of star tracker commands for stabilizing an inertially oriented satellite

Simple processors of star tracker commands for stabilizing inertially oriented satellit

Optimal uncertainty quantification for legacy data observations of Lipschitz functions

We consider the problem of providing optimal uncertainty quantification (UQ) --- and hence rigorous certification --- for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.Comment: 38 page

Visible and near infrared observation on the Global Aerosol Backscatter Experiment (GLOBE)

The Global Aerosol Backscatter Experiment (GLOBE) was intended to provide data on prevailing values of atmospheric backscatter cross-section. The primary intent was predicting the performance of spaceborne lidar systems, most notably the Laser Atmospheric Wind Sounder (LAWS) for the Earth Observing System (EOS). The second and related goal was to understand the source and characteristics of atmospheric aerosol particles. From the GLOBE flights, extensive data was obtained on the structure of clouds and the marine planetary boundary layer. A notable result for all observations is the consistency of the large increases in the aerosol scattering ratio for the marine boundary layer. Other results are noted

Spaces of finite element differential forms

We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit commuting projections. We present two families of spaces in the case of simplicial meshes, and two other families in the case of cubical meshes. We make use of the exterior calculus and the Koszul complex to define and understand the spaces. These tools allow us to treat a wide variety of situations, which are often treated separately, in a unified fashion.Comment: To appear in: Analysis and Numerics of Partial Differential Equations, U. Gianazza, F. Brezzi, P. Colli Franzone, and G. Gilardi, eds., Springer 2013. v2: a few minor typos corrected. v3: a few more typo correction

Weak Localization Thickness Measurements of Si:P Delta-Layers

We report on our results for the characterization of Si:P delta-layers grown by low temperature molecular beam epitaxy. Our data shows that the effective thickness of a delta-layer can be obtained through a weak localization analysis of electrical transport measurements performed in perpendicular and parallel magnetic fields. An estimate of the diffusivity of phosphorous in silicon is obtained by applying this method to several samples annealed at 850 Celsius for intervals of zero to 15 minutes. With further refinements, this may prove to be the most precise method of measuring delta-layer widths developed to date, including that of Secondary Ion Mass Spectrometry analysis

Aircraft motion and passenger comfort data from scheduled commercial airline flights

Data concerning the ride quality of aircraft taken on board commercial airline flights was presented. Five types of data are included: (1) root mean square (RMS) values of linear acceleration, angular acceleration or angular velocities, along with passenger subjective evaluations, (2) power spectra for the motion in each of six degrees of freedom, (3) scattergrams showing the probability density of the rms accelerations in the vertical and transverse directions, (4) probability distributions of the motion, and (5) on board noise levels during takeoff, climb, cruise, and descent