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Author: David Nicholas Gosset
Publisher:
ISBN:
Category :
Languages : en
Pages : 143
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Book Description
Quantum adiabatic optimization is a quantum algorithm for solving classical optimization problems (E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106). The solution to an optimization problem is encoded in the ground state of a "problem Hamiltonian" Hp which acts on the Hilbert space of n spin 1/2 particles and is diagonal in the Pauli z basis. To produce this ground state, one first initializes the quantum system in the ground state of a different Hamiltonian and then adiabatically changes the Hamiltonian into Hp. Farhi et al suggest the interpolating Hamiltonian [mathematical formula] ... where the parameter s is slowly changed as a function of time between 0 and 1. The running time of this algorithm is related to the minimum spectral gap of H(s) for s E (0, 11. We study such transverse field spin Hamiltonians using both analytic and numerical techniques. Our approach is example-based, that is, we study some specific choices for the problem Hamiltonian Hp which illustrate the breadth of phenomena which can occur. We present I A random ensemble of 3SAT instances which this algorithm does not solve efficiently. For these instances H(s) has a small eigenvalue gap at a value s* which approaches 1 as n - oc. II Theorems concerning the interpolating Hamiltonian when Hp is "scrambled" by conjugating with a random permutation matrix. III Results pertaining to phase transitions that occur as a function of the transverse field. IV A new quantum monte carlo method which can be used to compute ground state properties of such quantum systems. We discuss the implications of our results for the performance of quantum adiabatic optimization algorithms.
Author: David Nicholas Gosset
Publisher:
ISBN:
Category :
Languages : en
Pages : 143
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Book Description
Quantum adiabatic optimization is a quantum algorithm for solving classical optimization problems (E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution, 2000. arXiv:quant-ph/0001106). The solution to an optimization problem is encoded in the ground state of a "problem Hamiltonian" Hp which acts on the Hilbert space of n spin 1/2 particles and is diagonal in the Pauli z basis. To produce this ground state, one first initializes the quantum system in the ground state of a different Hamiltonian and then adiabatically changes the Hamiltonian into Hp. Farhi et al suggest the interpolating Hamiltonian [mathematical formula] ... where the parameter s is slowly changed as a function of time between 0 and 1. The running time of this algorithm is related to the minimum spectral gap of H(s) for s E (0, 11. We study such transverse field spin Hamiltonians using both analytic and numerical techniques. Our approach is example-based, that is, we study some specific choices for the problem Hamiltonian Hp which illustrate the breadth of phenomena which can occur. We present I A random ensemble of 3SAT instances which this algorithm does not solve efficiently. For these instances H(s) has a small eigenvalue gap at a value s* which approaches 1 as n - oc. II Theorems concerning the interpolating Hamiltonian when Hp is "scrambled" by conjugating with a random permutation matrix. III Results pertaining to phase transitions that occur as a function of the transverse field. IV A new quantum monte carlo method which can be used to compute ground state properties of such quantum systems. We discuss the implications of our results for the performance of quantum adiabatic optimization algorithms.
Author: William Cruz-Santos
Publisher: Springer Nature
ISBN: 3031025199
Category : Mathematics
Languages : en
Pages : 105
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Book Description
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
Author: N.B. Singh
Publisher: N.B. Singh
ISBN:
Category : Computers
Languages : en
Pages : 330
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Book Description
"Graph Theory: Adiabatic Quantum Computing Methods" explores the convergence of quantum computing and graph theory, offering a comprehensive examination of how quantum algorithms can tackle fundamental graph problems. From foundational concepts to advanced applications in fields like cryptography, machine learning, and network analysis, this book provides a clear pathway into the evolving landscape of quantum-enhanced graph algorithms. Designed for researchers, students, and professionals alike, it bridges theoretical insights with practical implementations, paving the way for innovative solutions in computational graph theory.
Author: Kai Liu
Publisher:
ISBN:
Category : Adiabatic invariants
Languages : en
Pages : 104
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Book Description
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.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
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Book Description
We studied the quantum adiabatic algorithm for combinatorial optimization. We obtained a simpler proof of the adiabatic theorem. However, we did not make much progress in developing new tools for rigorous analysis of the algorithm's performance on realistic optimization problems. This analysis seems to be substantially more difficult than for classical algorithms such as simulated annealing, because the ground state does not have a simple form. We also proved some interesting results about the complexity of the Local Consistency problem (deciding whether local density matrices that describe small pieces of a quantum system are consistent with a single overall state). In particular, we showed that this problem is QMA-complete. We also showed that N-representability, an important problem in quantum chemistry, is QMA-complete.
Author: Sebastian Feld
Publisher: Springer
ISBN: 3030140822
Category : Computers
Languages : en
Pages : 231
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Book Description
This book constitutes the refereed proceedings of the First International Workshop on Quantum Technology and Optimization Problems, QTOP 2019, held in Munich, Germany, in March 2019.The 18 full papers presented together with 1 keynote paper in this volume were carefully reviewed and selected from 21 submissions. The papers are grouped in the following topical sections: analysis of optimization problems; quantum gate algorithms; applications of quantum annealing; and foundations and quantum technologies.
Author: National Aeronautics and Space Adm Nasa
Publisher: Independently Published
ISBN: 9781723749322
Category : Science
Languages : en
Pages : 30
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Book Description
In this paper we analyze the performance of the Quantum Adiabatic Evolution (QAE) algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, gamma = M / N. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (verses only energy) is used, and are able to show the existence of a dynamic threshold gamma = gammad, beyond which QAE should take an exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz.Smelyanskiy, V. N. and Knysh, S. and Morris, R. D.Ames Research CenterALGORITHMS; COMBINATORIAL ANALYSIS; OPTIMIZATION; QUANTUM MECHANICS; ADIABATIC EQUATIONS; TOPOGRAPHY; EIGENVALUES; GRAPH THEORY; HAMILTONIAN FUNCTIONS; ANNEALING; APPROXIMATION; PROBABILITY THEORY
Author: Venkateswaran Kasirajan
Publisher: Springer Nature
ISBN: 3030636895
Category : Computers
Languages : en
Pages : 463
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Book Description
This introductory book on quantum computing includes an emphasis on the development of algorithms. Appropriate for both university students as well as software developers interested in programming a quantum computer, this practical approach to modern quantum computing takes the reader through the required background and up to the latest developments. Beginning with introductory chapters on the required math and quantum mechanics, Fundamentals of Quantum Computing proceeds to describe four leading qubit modalities and explains the core principles of quantum computing in detail. Providing a step-by-step derivation of math and source code, some of the well-known quantum algorithms are explained in simple ways so the reader can try them either on IBM Q or Microsoft QDK. The book also includes a chapter on adiabatic quantum computing and modern concepts such as topological quantum computing and surface codes. Features: o Foundational chapters that build the necessary background on math and quantum mechanics. o Examples and illustrations throughout provide a practical approach to quantum programming with end-of-chapter exercises. o Detailed treatment on four leading qubit modalities -- trapped-ion, superconducting transmons, topological qubits, and quantum dots -- teaches how qubits work so that readers can understand how quantum computers work under the hood and devise efficient algorithms and error correction codes. Also introduces protected qubits - 0-π qubits, fluxon parity protected qubits, and charge-parity protected qubits. o Principles of quantum computing, such as quantum superposition principle, quantum entanglement, quantum teleportation, no-cloning theorem, quantum parallelism, and quantum interference are explained in detail. A dedicated chapter on quantum algorithm explores both oracle-based, and Quantum Fourier Transform-based algorithms in detail with step-by-step math and working code that runs on IBM QisKit and Microsoft QDK. Topics on EPR Paradox, Quantum Key Distribution protocols, Density Matrix formalism, and Stabilizer formalism are intriguing. While focusing on the universal gate model of quantum computing, this book also introduces adiabatic quantum computing and quantum annealing. This book includes a section on fault-tolerant quantum computing to make the discussions complete. The topics on Quantum Error Correction, Surface codes such as Toric code and Planar code, and protected qubits help explain how fault tolerance can be built at the system level.
Author: Ronald C. Read
Publisher: Academic Press
ISBN: 1483263126
Category : Mathematics
Languages : en
Pages : 344
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Book Description
Graph Theory and Computing focuses on the processes, methodologies, problems, and approaches involved in graph theory and computer science. The book first elaborates on alternating chain methods, average height of planted plane trees, and numbering of a graph. Discussions focus on numbered graphs and difference sets, Euclidean models and complete graphs, classes and conditions for graceful graphs, and maximum matching problem. The manuscript then elaborates on the evolution of the path number of a graph, production of graphs by computer, and graph-theoretic programming language. Topics include FORTRAN characteristics of GTPL, design considerations, representation and identification of graphs in a computer, production of simple graphs and star topologies, and production of stars having a given topology. The manuscript examines the entropy of transformed finite-state automata and associated languages; counting hexagonal and triangular polyominoes; and symmetry of cubical and general polyominoes. Graph coloring algorithms, algebraic isomorphism invariants for graphs of automata, and coding of various kinds of unlabeled trees are also discussed. The publication is a valuable source of information for researchers interested in graph theory and computing.
Author: Catherine C. McGeoch
Publisher: Springer Nature
ISBN: 3031025180
Category : Mathematics
Languages : en
Pages : 83
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Book Description
Adiabatic quantum computation (AQC) is an alternative to the better-known gate model of quantum computation. The two models are polynomially equivalent, but otherwise quite dissimilar: one property that distinguishes AQC from the gate model is its analog nature. Quantum annealing (QA) describes a type of heuristic search algorithm that can be implemented to run in the ``native instruction set'' of an AQC platform. D-Wave Systems Inc. manufactures {quantum annealing processor chips} that exploit quantum properties to realize QA computations in hardware. The chips form the centerpiece of a novel computing platform designed to solve NP-hard optimization problems. Starting with a 16-qubit prototype announced in 2007, the company has launched and sold increasingly larger models: the 128-qubit D-Wave One system was announced in 2010 and the 512-qubit D-Wave Two system arrived on the scene in 2013. A 1,000-qubit model is expected to be available in 2014. This monograph presents an introductory overview of this unusual and rapidly developing approach to computation. We start with a survey of basic principles of quantum computation and what is known about the AQC model and the QA algorithm paradigm. Next we review the D-Wave technology stack and discuss some challenges to building and using quantum computing systems at a commercial scale. The last chapter reviews some experimental efforts to understand the properties and capabilities of these unusual platforms. The discussion throughout is aimed at an audience of computer scientists with little background in quantum computation or in physics. Table of Contents: Acknowledgments / Introduction / Adiabatic Quantum Computation / Quantum Annealing / The D-Wave Platform / Computational Experience / Bibliography / Author's Biography
