Programme Apollo | EPFL Graph Search

vignette|Lancement de la fusée de la mission . vignette|Le centre de contrôle de tir lors du lancement d. vignette|« C’est un petit pas pour [un] homme, mais un bond de géant pour l’Humanité » (Neil Armstrong - ).|alt= Le programme Apollo est le programme spatial de la NASA mené durant la période 1961 – 1972, qui a permis aux États-Unis d'envoyer pour la première fois des hommes sur la Lune. Il est lancé par le président John F. Kennedy le , essentiellement pour reconquérir le prestige américain mis à mal par les succès de l'astronautique soviétique, à une époque où la guerre froide entre les deux superpuissances battait son plein. Le programme avait pour objectif de poser un homme sur la Lune avant la fin de la décennie. Le , cet objectif était atteint par deux des trois membres d'équipage de la mission , Neil Armstrong et Buzz Aldrin. Cinq autres missions se sont posées par la suite sur d'autres sites lunaires et y ont séjourné jusqu'à trois jours. Ces expéditions ont permis de ra

TimeEvolver: A program for time evolution with improved error bound

Sebastian Zell

We present TimeEvolver, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix iH, where His the Hamiltonian, with an initial vector v. The fact that His Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation. Apart from effects of numerical roundoff, the resulting a posteriori error bound is rigorous, which represents a crucial novelty as compared to existing software packages such as Expokit[1]. On a standard notebook, TimeEvolverallows to compute time evolution with adjustable precision in Hilbert spaces of dimension greater than 10(6) Program summary Program Title: TimeEvolver CPC Library link to program files: https://doi.org/10.17632/vvwvng9w36.1 Code Ocean capsule: https://codeocean.com/capsule/8431379 Developer's repository link: https://github.com/marco-michel/TimeEvolver Licensing provisions: MIT Programming language: C++ Supplementary material: An example which demonstrates the computation of time evolution in a concrete physical system. Nature of problem: Computing time evolution in a generic physical quantum system can be reduced to the numerical task of calculating exp(-iHt)v. Here His the Hamiltonian matrix, which is large and sparse, icorresponds to the imaginary unit, tdenotes time and the vector vrepresents the initial state. A program is needed to perform this computation efficiently. Since the use of approximation methods is unavoidable, it is important to quantify as rigorously as possible the resulting error. Moreover, in order to facilitate the application to various problems in physics, additional functionalities are needed, in particular for forming the Hamiltonian matrix from a more abstract representation of the Hamiltonian operator. Solution method: The program employs known Krylov subspace methods for calculation the exponential of the large sparse matrix (- iHt) times the vector v. The Arnoldi algorithm is used to form the Krylov subspace and exponentiation of the resulting small matrix is achieved by diagonalization. The fact that (-iHt) is anti-Hermitian makes it possible to calculate the error of the Krylov approximation in terms of an easily-computable integral formula. This allows to choose a maximal size of the time step, after which the method is restarted and a new Krylov subspace is formed, while respecting an adjustable error bound. It is rigorous up to inaccuracies of a one-dimensional numerical integral and effects of finite machine precision, for which we also give an estimate. All linear algebra operations are performed with the Intel (R) Math Kernel Library and Boost is used for numerical integration. The methods for deriving the Hamiltonian matrix rely on a hashtable representation of Hilbert space. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).. Additionally, we provide routines for deriving the matrix Hfrom a more abstract representation of the Hamiltonian operator.

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TimeEvolver: A program for time evolution with improved error bound

Sebastian Zell

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