Download or read book Discrete Quantum Walks on Graphs and Digraphs written by Chris Godsil and published by Cambridge University Press. This book was released on 2022-12-31 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: Explore the mathematics arising from discrete quantum walks in this introduction to a rapidly developing area.
Download or read book Discrete Quantum Walks on Graphs and Digraphs written by Hanmeng Zhan and published by . This book was released on 2018 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach. A discrete quantum walk on a digraph $X$ is determined by a unitary matrix $U$, which acts on complex functions of the arcs of $X$. Generally speaking, $U$ is a product of two sparse unitary matrices, based on two direct-sum decompositions of the state space. Our goal is to relate properties of the walk to properties of $X$, given some of these decompositions. We start by exploring two models that involve coin operators, one due to Kendon, and the other due to Aharonov, Ambainis, Kempe, and Vazirani. While $U$ is not defined as a function in the adjacency matrix of the graph $X$, we find exact spectral correspondence between $U$ and $X$. This leads to characterization of rare phenomena, such as perfect state transfer and uniform average vertex mixing, in terms of the eigenvalues and eigenvectors of $X$. We also construct infinite families of graphs and digraphs that admit the aforementioned phenomena. The second part of this thesis analyzes abstract quantum walks, with no extra assumption on $U$. We show that knowing the spectral decomposition of $U$ leads to better understanding of the time-averaged limit of the probability distribution. In particular, we derive three upper bounds on the mixing time, and characterize different forms of uniform limiting distribution, using the spectral information of $U$. Finally, we construct a new model of discrete quantum walks from orientable embeddings of graphs. We show that the behavior of this walk largely depends on the vertex-face incidence structure. Circular embeddings of regular graphs for which $U$ has few eigenvalues are characterized. For instance, if $U$ has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric $2$-design, and $U$ is the exponential of a scalar multiple of the skew-symmetric adjacency matrix of an oriented graph. We prove that, for every regular embedding of a complete graph, $U$ is the transition matrix of a continuous quantum walk on an oriented graph.
Download or read book Discrete Quantum Walks on Graphs and Digraphs written by Chris Godsil and published by Cambridge University Press. This book was released on 2023-01-12 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
Download or read book Discrete Quantum Walks on Graphs and Digraphs written by Christopher David Godsil and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory"--
Download or read book Quantum Walks and Search Algorithms written by Renato Portugal and published by Springer. This book was released on 2018-08-20 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: The revised edition of this book offers an extended overview of quantum walks and explains their role in building quantum algorithms, in particular search algorithms. Updated throughout, the book focuses on core topics including Grover's algorithm and the most important quantum walk models, such as the coined, continuous-time, and Szedgedy's quantum walk models. There is a new chapter describing the staggered quantum walk model. The chapter on spatial search algorithms has been rewritten to offer a more comprehensive approach and a new chapter describing the element distinctness algorithm has been added. There is a new appendix on graph theory highlighting the importance of graph theory to quantum walks. As before, the reader will benefit from the pedagogical elements of the book, which include exercises and references to deepen the reader's understanding, and guidelines for the use of computer programs to simulate the evolution of quantum walks. Review of the first edition: “The book is nicely written, the concepts are introduced naturally, and many meaningful connections between them are highlighted. The author proposes a series of exercises that help the reader get some working experience with the presented concepts, facilitating a better understanding. Each chapter ends with a discussion of further references, pointing the reader to major results on the topics presented in the respective chapter.” - Florin Manea, zbMATH.
Download or read book Energy Minimization Methods in Computer Vision and Pattern Recognition written by Anand Rangarajan and published by Springer Science & Business Media. This book was released on 2005-10-31 with total page 671 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 5th International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, EMMCVPR 2005, held in St. Augustine, FL, USA in November 2005. The 24 revised full papers and 18 poster papers presented were carefully reviewed and selected from 120 submissions. The papers are organized in topical sections on probabilistic and informational approaches, combinatorial approaches, variational approaches, and other approaches and applications.
Download or read book Groups and Graphs Designs and Dynamics written by R. A. Bailey and published by Cambridge University Press. This book was released on 2024-05-30 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of four short courses looks at group representations, graph spectra, statistical optimality, and symbolic dynamics, highlighting their common roots in linear algebra. It leads students from the very beginnings in linear algebra to high-level applications: representations of finite groups, leading to probability models and harmonic analysis; eigenvalues of growing graphs from quantum probability techniques; statistical optimality of designs from Laplacian eigenvalues of graphs; and symbolic dynamics, applying matrix stability and K-theory. An invaluable resource for researchers and beginning Ph.D. students, this book includes copious exercises, notes, and references.
Download or read book Modern Trends in Algebra and Representation Theory written by David Jordan and published by Cambridge University Press. This book was released on 2023-08-17 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: Expanding upon the material delivered during the LMS Autumn Algebra School 2020, this volume reflects the fruitful connections between different aspects of representation theory. Each survey article addresses a specific subject from a modern angle, beginning with an exploration of the representation theory of associative algebras, followed by the coverage of important developments in Lie theory in the past two decades, before the final sections introduce the reader to three strikingly different aspects of group theory. Written at a level suitable for graduate students and researchers in related fields, this book provides pure mathematicians with a springboard into the vast and growing literature in each area.
Download or read book Algebraic Combinatorics and the Monster Group written by Alexander A. Ivanov and published by Cambridge University Press. This book was released on 2023-08-17 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering, arguably, one of the most attractive and mysterious mathematical objects, the Monster group, this text strives to provide an insightful introduction and the discusses the current state of the field. The Monster group is related to many areas of mathematics, as well as physics, from number theory to string theory. This book cuts through the complex nature of the field, highlighting some of the mysteries and intricate relationships involved. Containing many meaningful examples and a manual introduction to the computer package GAP, it provides the opportunity and resources for readers to start their own calculations. Some 20 experts here share their expertise spanning this exciting field, and the resulting volume is ideal for researchers and graduate students working in Combinatorial Algebra, Group theory and related areas.
Download or read book C Algebraic Geometry with Corners written by Kelli Francis-Staite and published by Cambridge University Press. This book was released on 2023-12-31 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: Schemes in algebraic geometry can have singular points, whereas differential geometers typically focus on manifolds which are nonsingular. However, there is a class of schemes, 'C∞-schemes', which allow differential geometers to study a huge range of singular spaces, including 'infinitesimals' and infinite-dimensional spaces. These are applied in synthetic differential geometry, and derived differential geometry, the study of 'derived manifolds'. Differential geometers also study manifolds with corners. The cube is a 3-dimensional manifold with corners, with boundary the six square faces. This book introduces 'C∞-schemes with corners', singular spaces in differential geometry with good notions of boundary and corners. They can be used to define 'derived manifolds with corners' and 'derived orbifolds with corners'. These have applications to major areas of symplectic geometry involving moduli spaces of J-holomorphic curves. This work will be a welcome source of information and inspiration for graduate students and researchers working in differential or algebraic geometry.
Download or read book Maurer Cartan Methods in Deformation Theory written by Vladimir Dotsenko and published by Cambridge University Press. This book was released on 2023-08-31 with total page 187 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering an exceptional range of topics, this text provides a unique overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.
Download or read book Surveys in Combinatorics 2024 written by Felix Fischer and published by Cambridge University Press. This book was released on 2024-06-13 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains surveys of current research directions in combinatorics written by leading researchers in their fields.
Download or read book The Calabi Problem for Fano Threefolds written by Carolina Araujo and published by Cambridge University Press. This book was released on 2023-06-30 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.
Download or read book Graph Based Representations in Pattern Recognition written by Luc Brun and published by Springer Science & Business Media. This book was released on 2005-03-23 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 5th IAPR International Workshop on Graph-Based Representations in Pattern Recognition, GbRPR 2005, held in Poitiers, France in April 2005. The 18 revised full papers and 17 revised poster papers presented were carefully reviewed and selected from 50 submissions. The papers are organized in topical sections on graph representations, graphs and linear representations, combinatorial maps, matching, hierarchical graph abstraction and matching, inexact
Download or read book Hypergraph Theory written by Alain Bretto and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.
Download or read book Topics in Topological Graph Theory written by Lowell W. Beineke and published by Cambridge University Press. This book was released on 2009-07-09 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references.
Download or read book The Nature of Computation written by Cristopher Moore and published by OUP Oxford. This book was released on 2011-08-11 with total page 1498 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again.
