Download or read book Solving the Bounded Depth Steiner Tree Problem Using an Adiabatic Quantum Computer written by Kai Liu and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Solving Optimization Problems Using Adiabatic Quantum Computing written by Kai Liu and published by . This book was released on 2018 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: The commercial D-Waves quantum annealer has given rise to plenty of interests due to the reported quantum speedup against classical annealing. In order to make use of this new technology, a problem must be formulated into a form of quadratic unconstrained binary optimization (QUBO) or Ising model. This thesis reports on case studies using a D-Wave quantum annealer to solve several optimization problems and providing results validation using classical exact approaches. In our thesis, we briefly introduce several classical techniques designed for QUBO problems and implement two exact approaches. With the proper tools, a D-Wave 2X computer consisted of 1098 active qubits is then evaluated for the Degree-Constrained Minimum Spanning Tree and the Steiner Tree problems, establishing their QUBO formulations are suitable for adiabatic quantum computers. Motivated by the remarkable performance, two more optimization problems are studied—the Bounded-Depth Steiner Tree problem and the Chromatic Sum problem. We propose a new formulation for each problem. The numbers of qubits (dimension of QUBO matrices) required by our formulations are O(|V|3) and O(|V|2) respectively, where |V| represents the number of vertices.

Download or read book Approximability of Optimization Problems through Adiabatic Quantum Computation written by William Cruz-Santos and published by Springer Nature. This book was released on 2022-05-31 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: The adiabatic quantum computation (AQC) is based on the adiabatic theorem to approximate solutions of the Schrödinger equation. The design of an AQC algorithm involves the construction of a Hamiltonian that describes the behavior of the quantum system. This Hamiltonian is expressed as a linear interpolation of an initial Hamiltonian whose ground state is easy to compute, and a final Hamiltonian whose ground state corresponds to the solution of a given combinatorial optimization problem. The adiabatic theorem asserts that if the time evolution of a quantum system described by a Hamiltonian is large enough, then the system remains close to its ground state. An AQC algorithm uses the adiabatic theorem to approximate the ground state of the final Hamiltonian that corresponds to the solution of the given optimization problem. In this book, we investigate the computational simulation of AQC algorithms applied to the MAX-SAT problem. A symbolic analysis of the AQC solution is given in order to understand the involved computational complexity of AQC algorithms. This approach can be extended to other combinatorial optimization problems and can be used for the classical simulation of an AQC algorithm where a Hamiltonian problem is constructed. This construction requires the computation of a sparse matrix of dimension 2n × 2n, by means of tensor products, where n is the dimension of the quantum system. Also, a general scheme to design AQC algorithms is proposed, based on a natural correspondence between optimization Boolean variables and quantum bits. Combinatorial graph problems are in correspondence with pseudo-Boolean maps that are reduced in polynomial time to quadratic maps. Finally, the relation among NP-hard problems is investigated, as well as its logical representability, and is applied to the design of AQC algorithms. It is shown that every monadic second-order logic (MSOL) expression has associated pseudo-Boolean maps that can be obtained by expanding the given expression, and also can be reduced to quadratic forms. Table of Contents: Preface / Acknowledgments / Introduction / Approximability of NP-hard Problems / Adiabatic Quantum Computing / Efficient Hamiltonian Construction / AQC for Pseudo-Boolean Optimization / A General Strategy to Solve NP-Hard Problems / Conclusions / Bibliography / Authors' Biographies

Download or read book The Steiner Tree Problem written by F.K. Hwang and published by Elsevier. This book was released on 1992-10-20 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Steiner problem asks for a shortest network which spans a given set of points. Minimum spanning networks have been well-studied when all connections are required to be between the given points. The novelty of the Steiner tree problem is that new auxiliary points can be introduced between the original points so that a spanning network of all the points will be shorter than otherwise possible. These new points are called Steiner points - locating them has proved problematic and research has diverged along many different avenues.This volume is devoted to the assimilation of the rich field of intriguing analyses and the consolidation of the fragments. A section has been given to each of the three major areas of interest which have emerged. The first concerns the Euclidean Steiner Problem, historically the original Steiner tree problem proposed by Jarník and Kössler in 1934. The second deals with the Steiner Problem in Networks, which was propounded independently by Hakimi and Levin and has enjoyed the most prolific research amongst the three areas. The Rectilinear Steiner Problem, introduced by Hanan in 1965, is discussed in the third part. Additionally, a forth section has been included, with chapters discussing areas where the body of results is still emerging.The collaboration of three authors with different styles and outlooks affords individual insights within a cohesive whole.

Download or read book Advances in Steiner Trees written by Ding-Zhu Du and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Volume on Advances in Steiner Trees is divided into two sections. The first section of the book includes papers on the general geometric Steiner tree problem in the plane and higher dimensions. The second section of the book includes papers on the Steiner problem on graphs. The general geometric Steiner tree problem assumes that you have a given set of points in some d-dimensional space and you wish to connect the given points with the shortest network possible. The given set ofpoints are 3 Figure 1: Euclidean Steiner Problem in E usually referred to as terminals and the set ofpoints that may be added to reduce the overall length of the network are referred to as Steiner points. What makes the problem difficult is that we do not know a priori the location and cardinality ofthe number ofSteiner points. Thus)the problem on the Euclidean metric is not known to be in NP and has not been shown to be NP-Complete. It is thus a very difficult NP-Hard problem.

Download or read book Steiner Tree Problems in Computer Communication Networks written by Dingzhu Du and published by World Scientific. This book was released on 2008-01-01 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Steiner tree problem is one of the most important combinatorial optimization problems. It has a long history that can be traced back to the famous mathematician Fermat (1601-1665). This book studies three significant breakthroughs on the Steiner tree problem that were achieved in the 1990s, and some important applications of Steiner tree problems in computer communication networks researched in the past fifteen years. It not only covers some of the most recent developments in Steiner tree problems, but also discusses various combinatorial optimization methods, thus providing a balance between theory and practice. Sample Chapter(s). Chapter 1: Minimax Approach and Steiner Ratio (372 KB). Contents: Minimax Approach and Steiner Ratio; k -Steiner Ratios and Better Approximation Algorithms; Geometric Partitions and Polynomial Time Approximation Schemes; Grade of Service Steiner Tree Problem; Steiner Tree Problem for Minimal Steiner Points; Bottleneck Steiner Tree Problem; Steiner k -Tree and k -Path Routing Problems; Steiner Tree Coloring Problem; Steiner Tree Scheduling Problem; Survivable Steiner Network Problem. Readership: Researchers and graduate students of computer science and engineering as well as operations research.

Download or read book Approximate Solutions to the Steiner Tree Problem written by Albert Thomas Borchers and published by . This book was released on 1996 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book An Iterative Approach for Delay bounded Minimum Steiner Tree Construction written by Jing Zhu and published by . This book was released on 1994 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Quantum State Transfer and Network Engineering written by Georgios M. Nikolopoulos and published by Springer Science & Business Media. This book was released on 2013-10-05 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: Faithful communication is a necessary precondition for large-scale quantum information processing and networking, irrespective of the physical platform. Thus, the problems of quantum-state transfer and quantum-network engineering have attracted enormous interest over the last years, and constitute one of the most active areas of research in quantum information processing. The present volume introduces the reader to fundamental concepts and various aspects of this exciting research area, including links to other related areas and problems. The implementation of state-transfer schemes and the engineering of quantum networks are discussed in the framework of various quantum optical and condensed matter systems, emphasizing the interdisciplinary character of the research area. Each chapter is a review of theoretical or experimental achievements on a particular topic, written by leading scientists in the field. The volume aims at both newcomers as well as experienced researchers.

Download or read book Quantum Information Processing and Quantum Error Correction written by Ivan Djordjevic and published by Academic Press. This book was released on 2012-04-16 with total page 597 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum Information Processing and Quantum Error Correction is a self-contained, tutorial-based introduction to quantum information, quantum computation, and quantum error-correction. Assuming no knowledge of quantum mechanics and written at an intuitive level suitable for the engineer, the book gives all the essential principles needed to design and implement quantum electronic and photonic circuits. Numerous examples from a wide area of application are given to show how the principles can be implemented in practice. This book is ideal for the electronics, photonics and computer engineer who requires an easy- to-understand foundation on the principles of quantum information processing and quantum error correction, together with insight into how to develop quantum electronic and photonic circuits. Readers of this book will be ready for further study in this area, and will be prepared to perform independent research. The reader completed the book will be able design the information processing circuits, stabilizer codes, Calderbank-Shor-Steane (CSS) codes, subsystem codes, topological codes and entanglement-assisted quantum error correction codes; and propose corresponding physical implementation. The reader completed the book will be proficient in quantum fault-tolerant design as well. Unique Features Unique in covering both quantum information processing and quantum error correction - everything in one book that an engineer needs to understand and implement quantum-level circuits. Gives an intuitive understanding by not assuming knowledge of quantum mechanics, thereby avoiding heavy mathematics. In-depth coverage of the design and implementation of quantum information processing and quantum error correction circuits. Provides the right balance among the quantum mechanics, quantum error correction, quantum computing and quantum communication. Dr. Djordjevic is an Assistant Professor in the Department of Electrical and Computer Engineering of College of Engineering, University of Arizona, with a joint appointment in the College of Optical Sciences. Prior to this appointment in August 2006, he was with University of Arizona, Tucson, USA (as a Research Assistant Professor); University of the West of England, Bristol, UK; University of Bristol, Bristol, UK; Tyco Telecommunications, Eatontown, USA; and National Technical University of Athens, Athens, Greece. His current research interests include optical networks, error control coding, constrained coding, coded modulation, turbo equalization, OFDM applications, and quantum error correction. He presently directs the Optical Communications Systems Laboratory (OCSL) within the ECE Department at the University of Arizona. Provides everything an engineer needs in one tutorial-based introduction to understand and implement quantum-level circuits Avoids the heavy use of mathematics by not assuming the previous knowledge of quantum mechanics Provides in-depth coverage of the design and implementation of quantum information processing and quantum error correction circuits

Download or read book Phase Transitions in Combinatorial Optimization Problems written by Alexander K. Hartmann and published by John Wiley & Sons. This book was released on 2006-05-12 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: A concise, comprehensive introduction to the topic of statistical physics of combinatorial optimization, bringing together theoretical concepts and algorithms from computer science with analytical methods from physics. The result bridges the gap between statistical physics and combinatorial optimization, investigating problems taken from theoretical computing, such as the vertex-cover problem, with the concepts and methods of theoretical physics. The authors cover rapid developments and analytical methods that are both extremely complex and spread by word-of-mouth, providing all the necessary basics in required detail. Throughout, the algorithms are shown with examples and calculations, while the proofs are given in a way suitable for graduate students, post-docs, and researchers. Ideal for newcomers to this young, multidisciplinary field.

Download or read book Mathematical Reviews written by and published by . This book was released on 2005 with total page 836 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Quantum Computer Science written by N. David Mermin and published by Cambridge University Press. This book was released on 2007-08-30 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the 1990's it was realized that quantum physics has some spectacular applications in computer science. This book is a concise introduction to quantum computation, developing the basic elements of this new branch of computational theory without assuming any background in physics. It begins with an introduction to the quantum theory from a computer-science perspective. It illustrates the quantum-computational approach with several elementary examples of quantum speed-up, before moving to the major applications: Shor's factoring algorithm, Grover's search algorithm, and quantum error correction. The book is intended primarily for computer scientists who know nothing about quantum theory, but will also be of interest to physicists who want to learn the theory of quantum computation, and philosophers of science interested in quantum foundational issues. It evolved during six years of teaching the subject to undergraduates and graduate students in computer science, mathematics, engineering, and physics, at Cornell University.

Download or read book Deterministic Operations Research written by David J. Rader and published by John Wiley & Sons. This book was released on 2013-06-07 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uniquely blends mathematical theory and algorithm design for understanding and modeling real-world problems Optimization modeling and algorithms are key components to problem-solving across various fields of research, from operations research and mathematics to computer science and engineering. Addressing the importance of the algorithm design process. Deterministic Operations Research focuses on the design of solution methods for both continuous and discrete linear optimization problems. The result is a clear-cut resource for understanding three cornerstones of deterministic operations research: modeling real-world problems as linear optimization problem; designing the necessary algorithms to solve these problems; and using mathematical theory to justify algorithmic development. Treating real-world examples as mathematical problems, the author begins with an introduction to operations research and optimization modeling that includes applications form sports scheduling an the airline industry. Subsequent chapters discuss algorithm design for continuous linear optimization problems, covering topics such as convexity. Farkas’ Lemma, and the study of polyhedral before culminating in a discussion of the Simplex Method. The book also addresses linear programming duality theory and its use in algorithm design as well as the Dual Simplex Method. Dantzig-Wolfe decomposition, and a primal-dual interior point algorithm. The final chapters present network optimization and integer programming problems, highlighting various specialized topics including label-correcting algorithms for the shortest path problem, preprocessing and probing in integer programming, lifting of valid inequalities, and branch and cut algorithms. Concepts and approaches are introduced by outlining examples that demonstrate and motivate theoretical concepts. The accessible presentation of advanced ideas makes core aspects easy to understand and encourages readers to understand how to think about the problem, not just what to think. Relevant historical summaries can be found throughout the book, and each chapter is designed as the continuation of the “story” of how to both model and solve optimization problems by using the specific problems-linear and integer programs-as guides. The book’s various examples are accompanied by the appropriate models and calculations, and a related Web site features these models along with MapleTM and MATLAB® content for the discussed calculations. Thoroughly class-tested to ensure a straightforward, hands-on approach, Deterministic Operations Research is an excellent book for operations research of linear optimization courses at the upper-undergraduate and graduate levels. It also serves as an insightful reference for individuals working in the fields of mathematics, engineering, computer science, and operations research who use and design algorithms to solve problem in their everyday work.

Download or read book A Mathematical Introduction to Electronic Structure Theory written by Lin Lin and published by SIAM. This book was released on 2019-06-05 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on first principle quantum mechanics, electronic structure theory is widely used in physics, chemistry, materials science, and related fields and has recently received increasing research attention in applied and computational mathematics. This book provides a self-contained, mathematically oriented introduction to the subject and its associated algorithms and analysis. It will help applied mathematics students and researchers with minimal background in physics understand the basics of electronic structure theory and prepare them to conduct research in this area. The book begins with an elementary introduction of quantum mechanics, including the uncertainty principle and the Hartree?Fock theory, which is considered the starting point of modern electronic structure theory. The authors then provide an in-depth discussion of two carefully selected topics that are directly related to several aspects of modern electronic structure calculations: density matrix based algorithms and linear response theory. Chapter 2 introduces the Kohn?Sham density functional theory with a focus on the density matrix based numerical algorithms, and Chapter 3 introduces linear response theory, which provides a unified viewpoint of several important phenomena in physics and numerics. An understanding of these topics will prepare readers for more advanced topics in this field. The book concludes with the random phase approximation to the correlation energy. The book is written for advanced undergraduate and beginning graduate students, specifically those with mathematical backgrounds but without a priori knowledge of quantum mechanics, and can be used for self-study by researchers, instructors, and other scientists. The book can also serve as a starting point to learn about many-body perturbation theory, a topic at the frontier of the study of interacting electrons.

Download or read book Complexity and Real Computation written by Lenore Blum and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.

Download or read book Mathematical Analysis of Evolution Information and Complexity written by Wolfgang Arendt and published by John Wiley & Sons. This book was released on 2009-07-10 with total page 502 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a wide collection of research in mathematics, information theory, physics and other scientific and technical areas, this new title offers elementary and thus easily accessible introductions to the various fields of research addressed in the book.