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Author: Chris Godsil Publisher: Cambridge University Press ISBN: 1009261703 Category : Mathematics Languages : en Pages : 152
Book Description
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.
Author: Hanmeng Zhan Publisher: ISBN: Category : Algebraic topology Languages : en Pages : 144
Book Description
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.
Author: Christopher David Godsil Publisher: ISBN: 9781009261692 Category : Algorithms Languages : en Pages : 0
Book Description
"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"--
Author: Renato Portugal Publisher: Springer ISBN: 3319978136 Category : Science Languages : en Pages : 314
Book Description
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.
Author: Anand Rangarajan Publisher: Springer Science & Business Media ISBN: 3540302875 Category : Computers Languages : en Pages : 671
Book Description
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.
Author: R. A. Bailey Publisher: Cambridge University Press ISBN: 1009465945 Category : Mathematics Languages : en Pages : 452
Book Description
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.
Author: David Jordan Publisher: Cambridge University Press ISBN: 1009097350 Category : Mathematics Languages : en Pages : 407
Book Description
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.
Author: Alexander A. Ivanov Publisher: Cambridge University Press ISBN: 1009338056 Category : Mathematics Languages : en Pages : 584
Book Description
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.
Author: Kelli Francis-Staite Publisher: Cambridge University Press ISBN: 1009400207 Category : Mathematics Languages : en Pages : 224
Book Description
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.
Author: Chris Godsil
Author: Chris Godsil
Author: Chris Godsil
Author: Hanmeng Zhan
Author: Christopher David Godsil
Author: Renato Portugal
Author: Anand Rangarajan
Author: R. A. Bailey
Author: David Jordan
Author: Alexander A. Ivanov
Author: Kelli Francis-Staite