Introduction to topological quantum computation with non-Abelian anyons

Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.Comment: 51 pages, 51 figure

Optimized Periodic Coulomb Potential in Two Dimension

The 1/r Coulomb potential is calculated for a two dimensional system with periodic boundary conditions. Using polynomial splines in real space and a summation in reciprocal space we obtain numerically optimized potentials which allow us efficient calculations of any periodic (long-ranged) potential up to high precision. We discuss the parameter space of the optimized potential for the periodic Coulomb potential. Compared to the analytic Ewald potential, the optimized potentials can reach higher precisions by up to several orders of magnitude. We explicitly give simple expressions for fast calculations of the periodic Coulomb potential where the summation in reciprocal space is reduced to a few terms

A Spline LR Test for Goodness-of-Fit

Goodness-of-Fit tests, nuisance parameters, cubic spline, Neyman smooth test, Lagrange Multiplier test, stable distributions, student t distributions

Thermal Emission of WASP-14b Revealed with Three Spitzer Eclipses

Exoplanet WASP-14b is a highly irradiated, transiting hot Jupiter. Joshi et al. calculate an equilibrium temperature Teq of 1866 K for zero albedo and reemission from the entire planet, a mass of 7.3 +/- 0.5 Jupiter masses and a radius of 1.28 +/- 0.08 Jupiter radii. Its mean density of 4.6 g/cm3 is one of the highest known for planets with periods less than 3 days. We obtained three secondary eclipse light curves with the Spitzer Space Telescope. The eclipse depths from the best jointly fit model are $0.224\%$ +/- $0.018\%$ at 4.5 {\mu}m and $0.181\%$ +/- $0.022\%$ at 8.0 {\mu}m. The corresponding brightness temperatures are 2212 +/- 94 K and 1590 +/- 116 K. A slight ambiguity between systematic models suggests a conservative 3.6 {\mu}m eclipse depth of $0.19\%$ +/- $0.01\%$ and brightness temperature of 2242 +/- 55 K. Although extremely irradiated, WASP-14b does not show any distinct evidence of a thermal inversion. In addition, the present data nominally favor models with day night energy redistribution less than $~30\%$. The current data are generally consistent with oxygen-rich as well as carbon-rich compositions, although an oxygen-rich composition provides a marginally better fit. We confirm a significant eccentricity of e = 0.087 +/- 0.002 and refine other orbital parameters.Comment: 16 pages, 16 figure

Multivariate Splines and Algebraic Geometry

Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems

Broadband near-infrared astronomical spectrometer calibration and on-sky validation with an electro-optic laser frequency comb

The quest for extrasolar planets and their characterisation as well as studies of fundamental physics on cosmological scales rely on capabilities of high-resolution astronomical spectroscopy. A central requirement is a precise wavelength calibration of astronomical spectrographs allowing for extraction of subtle wavelength shifts from the spectra of stars and quasars. Here, we present an all-fibre, 400 nm wide near-infrared frequency comb based on electro-optic modulation with 14.5 GHz comb line spacing. Tests on the high-resolution, near-infrared spectrometer GIANO-B show a photon-noise limited calibration precision of <10 cm/s as required for Earth-like planet detection. Moreover, the presented comb provides detailed insight into particularities of the spectrograph such as detector inhomogeneities and differential spectrograph drifts. The system is validated in on-sky observations of a radial velocity standard star (HD221354) and telluric atmospheric absorption features. The advantages of the system include simplicity, robustness and turn-key operation, features that are valuable at the observation sites

Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of "arithmetic TQFT" and conjecture (with supporting numerical evidence) that SL(2,C) Chern-Simons theory is an example of such a theory.Comment: 60 pages, 9 figure