Quadrature formulas based on rational interpolation

We consider quadrature formulas based on interpolation using the basis functions $1/(1+t_kx)$ $(k=1,2,3,\ldots)$ on $[-1,1]$, where $t_k$ are parameters on the interval $(-1,1)$. We investigate two types of quadratures: quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible (Gaussian quadrature), and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions (orthogonal quadrature). We show that both approaches involve orthogonal polynomials with modified (or varying) weights which depend on the number of quadrature nodes. The asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes

Temporal-mode continuous-variable cluster states using linear optics

I present an extensible experimental design for optical continuous-variable cluster states of arbitrary size using four offline (vacuum) squeezers and six beamsplitters. This method has all the advantages of a temporal-mode encoding [Phys. Rev. Lett. 104, 250503], including finite requirements for coherence and stability even as the computation length increases indefinitely, with none of the difficulty of inline squeezing. The extensibility stems from a construction based on Gaussian projected entangled pair states (GPEPS). The potential for use of this design within a fully fault tolerant model is discussed.Comment: 9 pages, 19 color figure

On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented via nonoscillatory phase functions, even in the high-frequency regime. Our algorithm achieves near machine precision accuracy and the time required to compute one root of a solution is independent of the frequency of oscillations of that solution. Moreover, despite its great generality, our approach is competitive with specialized, state-of-the-art methods for the construction of Gaussian quadrature rules of large orders when it used in such a capacity. The performance of the scheme is illustrated with several numerical experiments and a Fortran implementation of our algorithm is available at the author's website