27,525 research outputs found

    Quadratic Form Expansions for Unitaries

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    We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over R\mathbb R. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various conditions under which it is possible to efficiently implement a unitary operation U, either when provided a quadratic form expansion for U as input, or by finding a quadratic form expansion for U from other input data.Comment: 20 pages, 3 figures; (extended version of) accepted submission to TQC 200

    Probing Thermalization through Spectral Analysis with Matrix Product Operators

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    We combine matrix product operator techniques with Chebyshev polynomial expansions and present a method that is able to explore spectral properties of quantum many-body Hamiltonians. In particular, we show how this method can be used to probe thermalization of large spin chains without explicitly simulating their time evolution, as well as to compute full and local densities of states. The performance is illustrated with the examples of the Ising and PXP spin chains. For the nonintegrable Ising chain, our findings corroborate the presence of thermalization for several initial states, well beyond what direct timedependent simulations have been able to achieve so far

    Diffuse interface models of locally inextensible vesicles in a viscous fluid

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    We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. To solve for the local Lagrange multiplier, we introduce a new equation whose solution essentially provides a harmonic extension of the local Lagrange multiplier off the interface while maintaining the local inextensibility constraint near the interface. To make the method more robust, we develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. Asymptotic analysis is presented that shows that our new system converges to a relaxed version of the inextensible sharp interface model. This is also verified numerically. Although the model does not depend on dimension, we present numerical simulations only in 2D. To solve the 2D equations numerically, we develop an efficient algorithm combining an operator splitting approach with adaptive finite elements where the Navier-Stokes equations are implicitly coupled to the diffuse interface inextensibility equation. Numerical simulations of a single vesicle in a shear flow at different Reynolds numbers demonstrate that errors in enforcing local inextensibility may accumulate and lead to large differences in the dynamics in the tumbling regime and differences in the inclination angle of vesicles in the tank-treading regime. The local relaxation algorithm is shown to effectively prevent this accumulation by driving the system back to its equilibrium state when errors in local inextensibility arise.Comment: 25 page

    Approximating the Probability Distribution of Functions of Random Variables: A New Approach

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    We introduce a new approximation method for the distribution of functions of random variables that are real-valued. The approximation involves moment matching and exploits properties of the class of normal inverse Gaussian distributions. In the paper we examine the how well the different approximation methods can capture the tail behavior of a function of random variables relative each other. This is done by simulate a number functions of random variables and then investigate the tail behavior for each method. Further we also focus on the regions of unimodality and positive definiteness of the different approximation methods. We show that the new method provides equal or better approximations than Gram-Charlier and Edgeworth expansions. Nous introduisons une nouvelle méthode pour approximer la distribution de variables aléatoires. L'approximation est basée sur la classe de distribution normale inverse gaussienne. On démontre que la nouvelle approximation est meilleure que les expansions Gram-Charlier et Edgeworth.normal inverse Gaussian, Edgeworth expansions, Gram-Charlier, distribution normale inverse gaussienne, expansions d'Edgeworth, Gram-Charlier

    Approximating the probability distribution of functions of random variables: A new approach

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    We introduce a new approximation method for the distribution of functions of random variables that are real-valued. The approximation involves moment matching and exploits properties of the class of normal inverse Gaussian distributions. In the paper we we examine the how well the different approximation methods can capture the tail behavior of a function of random variables relative each other. This is obtain done by simulate a number functions of random variables and then investigate the tail behavior for each method. Further we also focus on the regions of unimodality and positive definiteness of the different approximation methods. We show that the new method provides equal or better approximations than Gram-Charlier and Edgeworth expansioApproximation of random variables
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