19 research outputs found

    Constructing networks of quantum channels for state preparation

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    Entangled possibly mixed states are an essential resource for quantum computation, communication, metrology, and the simulation of many-body systems. It is important to develop and improve preparation protocols for such states. One possible way to prepare states of interest is to design an open system that evolves only towards the desired states. A Markovian evolution of a quantum system can be generally described by a Lindbladian. Tensor networks provide a framework to construct physically relevant entangled states. In particular, matrix product density operators (MPDOs) form an important variational class of states. MPDOs generalize matrix product states to mixed states, can represent thermal states of local one-dimensional Hamiltonians at sufficiently large temperatures, describe systems that satisfy the area law of entanglement, and form the basis of powerful numerical methods. In this work we develop an algorithm that determines for a given linear subspace of MPDOs whether this subspace can be the stable space of some frustration free k-local Lindbladian and, if so, outputs an appropriate Lindbladian. We proceed by using machine learning with networks of quantum channels, also known as quantum neural networks (QNNs), to train denoising post-processing devices for quantum sources. First, we show that QNNs can be trained on imperfect devices even when part of the training data is corrupted. Second, we show that QNNs can be trained to extrapolate quantum states to, e.g., lower temperatures. Third, we show how to denoise quantum states in an unsupervised manner. We develop a novel quantum autoencoder that successfully denoises Greenberger-Horne-Zeilinger, W, Dicke, and cluster states subject to spin-flip, dephasing errors, and random unitary noise. Finally, we develop recurrent QNNs (RQNNs) for denoising that requires memory, such as combating drifts. RQNNs can be thought of as matrix product quantum channels with a quantum algorithm for training and are closely related to MPDOs. The proposed preparation and denoising protocols can be beneficial for various emergent quantum technologies and are within reach of present-day experiments

    MATLAB

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    A well-known statement says that the PID controller is the "bread and butter" of the control engineer. This is indeed true, from a scientific standpoint. However, nowadays, in the era of computer science, when the paper and pencil have been replaced by the keyboard and the display of computers, one may equally say that MATLAB is the "bread" in the above statement. MATLAB has became a de facto tool for the modern system engineer. This book is written for both engineering students, as well as for practicing engineers. The wide range of applications in which MATLAB is the working framework, shows that it is a powerful, comprehensive and easy-to-use environment for performing technical computations. The book includes various excellent applications in which MATLAB is employed: from pure algebraic computations to data acquisition in real-life experiments, from control strategies to image processing algorithms, from graphical user interface design for educational purposes to Simulink embedded systems

    Fast integer multiplication using generalized Fermat primes

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    For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n ×\times log n ×\times log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer proved that there exists K > 1 and an algorithm performing this operation in O(n ×\times log n ×\times K log n). Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get K = 8, and conjecturally K = 4. Using an alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, we obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model

    On Flows, Paths, Roots, and Zeros

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    This thesis has two parts; in the first of which we give new results for various network flow problems. (1) We present a novel dual ascent algorithm for min-cost flow and show that an implementation of it is very efficient on certain instance classes. (2) We approach the problem of numerical stability of interior point network flow algorithms by giving a path following method that works with integer arithmetic solely and is thus guaranteed to be free of any nu-merical instabilities. (3) We present a gradient descent approach for the undirected transship-ment problem and its special case, the single source shortest path problem (SSSP). For distrib-uted computation models this yields the first SSSP-algorithm with near-optimal number of communication rounds. The second part deals with fundamental topics from algebraic computation. (1) We give an algorithm for computing the complex roots of a complex polynomial. While achieving a com-parable bit complexity as previous best results, our algorithm is simple and promising to be of practical impact. It uses a test for counting the roots of a polynomial in a region that is based on Pellet's theorem. (2) We extend this test to polynomial systems, i.e., we develop an algorithm that can certify the existence of a k-fold zero of a zero-dimensional polynomial system within a given region. For bivariate systems, we show experimentally that this approach yields signifi-cant improvements when used as inclusion predicate in an elimination method.Im ersten Teil dieser Dissertation präsentieren wir neue Resultate für verschiedene Netzwerkflussprobleme. (1)Wir geben eine neue Duale-Aufstiegsmethode für das Min-Cost-Flow- Problem an und zeigen, dass eine Implementierung dieser Methode sehr effizient auf gewissen Instanzklassen ist. (2)Wir behandeln numerische Stabilität von Innere-Punkte-Methoden fürNetwerkflüsse, indem wir eine solche Methode angeben die mit ganzzahliger Arithmetik arbeitet und daher garantiert frei von numerischen Instabilitäten ist. (3) Wir präsentieren ein Gradienten-Abstiegsverfahren für das ungerichtete Transshipment-Problem, und seinen Spezialfall, das Single-Source-Shortest-Problem (SSSP), die für SSSP in verteilten Rechenmodellen die erste mit nahe-optimaler Anzahl von Kommunikationsrunden ist. Der zweite Teil handelt von fundamentalen Problemen der Computeralgebra. (1) Wir geben einen Algorithmus zum Berechnen der komplexen Nullstellen eines komplexen Polynoms an, der eine vergleichbare Bitkomplexität zu vorherigen besten Resultaten hat, aber vergleichsweise einfach und daher vielversprechend für die Praxis ist. (2)Wir erweitern den darin verwendeten Pellet-Test zum Zählen der Nullstellen eines Polynoms auf Polynomsysteme, sodass wir die Existenz einer k-fachen Nullstelle eines Systems in einer gegebenen Region zertifizieren können. Für bivariate Systeme zeigen wir experimentell, dass eine Integration dieses Ansatzes in eine Eliminationsmethode zu einer signifikanten Verbesserung führt

    The Craft of Economic Modeling

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    This book is a practical guide to building economic models, both macroeconomic and multisectoral. It uses free software available from the Internet together with regularly updated databanks including the quarterly national accounts of the United States. It does NOT deal with some recent fads in economic model building, such as Real Business Cycles, Computable General Equilibriium models, nor DSGE models, all of which are, in the author's opinion, "alternative reality" models at best

    Nearly optimal computations with structured matrices

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    International audienceWe estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation. We supply them now as well as the Boolean complexity estimates for the important problems of multiplication of transposed Vandermonde matrix and its inverse by a vector. All known Boolean cost estimates from [10] for such problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer’s important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes, as well as the transposed Vandermonde matrices. It is known that the solution of Toeplitz, Hankel, Cauchy, Vandermonde, and transposed Vandermonde linear systems of equations is generally prone to numerical stability problems, and numerical problems arise even for multiplication of Cauchy, Vandermonde, and transposed Vandermonde matrices by a vector. Thus our FFT-based results on the Boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices, showing rather moderate growth of the complexity as the input size increases

    Nearly Optimal Refinement of Real Roots of a Univariate Polynomial

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    International audienceWe assume that a real square-free polynomial AA has a degree dd, a maximum coefficient bitsize τ\tau and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then we combine the {\em Double Exponential Sieve} algorithm (also called the {\em Bisection of the Exponents}), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of t=2−Lt=2^{-L}. The algorithm has Boolean complexity OB (d2τ+dL)O_B~(d^2 \tau + d L ). This substantially decreases the known bound OB (d3+d2L)O_B~(d^3 +d^2L) and is optimal up to a polylogarithmic factor. Furthermore we readily extend our algorithm to support the same upper bound on the complexity of the refinement of rr real roots, for any r≤dr\le d, by incorporating the known efficient algorithms for multipoint polynomial evaluation. The main ingredient for the latter is an efficient algorithm for (approximate) polynomial division; we present a variation based on structured matrix computation with quasi-optimal Boolean complexity

    Nearly Optimal Computations with Structured Matrices

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    We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic most popular classes, that is, Toeplitz, Hankel, Cauchy and Van-der-monde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we supply now. All known Boolean cost estimates for these problems rely on using Kronecker product. This implies the dd-fold precision increase for the dd-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representation of our tasks and algorithms in terms of both structured matrices and polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes.Comment: (2014-04-10
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