Lagrangian Duality and Adiabatic Quantum Computation for Constrained Optimization Problems PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Lagrangian Duality and Adiabatic Quantum Computation for Constrained Optimization Problems PDF full book. Access full book title Lagrangian Duality and Adiabatic Quantum Computation for Constrained Optimization Problems by Einar Gabbassov. Download full books in PDF and EPUB format.
Author: Einar Gabbassov
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
The Quantum Approximate Optimization Algorithm (QAOA) is a heuristic method for solving unconstrained binary optimization problems with a gate-based quantum computer. The QAOA consists of a particular quantum circuit architecture, together with a prescription for choosing the parameterization of the circuit. The first part of the thesis studies both the architecture and optimal parameterization of the QAOA circuit. After reviewing the necessary mathematical and physical background, we derive QAOA from scratch and discuss some of its properties. The second part of the thesis focuses on solving constrained combinatorial optimization problems in the setting of fault-tolerant quantum computation and presents a novel Lagrangian duality approach to Discretized Adiabatic Quantum Computation (DAQC). The proposed method allows for building highly resource-efficient and parallelizable quantum circuits. The thesis presents numerical evidence that demonstrates that the proposed approach gives the quadratic improvement in circuit complexity and evolution time over circuits derived from the traditional Quadratic Unconstrained Binary Optimization (QUBO) formalism. We illustrate our findings in the benchmark of the QUBO- and Lagrangian-based DAQC on the NP-complete 1D 0-1 knapsack problem.
Author: Einar Gabbassov
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
The Quantum Approximate Optimization Algorithm (QAOA) is a heuristic method for solving unconstrained binary optimization problems with a gate-based quantum computer. The QAOA consists of a particular quantum circuit architecture, together with a prescription for choosing the parameterization of the circuit. The first part of the thesis studies both the architecture and optimal parameterization of the QAOA circuit. After reviewing the necessary mathematical and physical background, we derive QAOA from scratch and discuss some of its properties. The second part of the thesis focuses on solving constrained combinatorial optimization problems in the setting of fault-tolerant quantum computation and presents a novel Lagrangian duality approach to Discretized Adiabatic Quantum Computation (DAQC). The proposed method allows for building highly resource-efficient and parallelizable quantum circuits. The thesis presents numerical evidence that demonstrates that the proposed approach gives the quadratic improvement in circuit complexity and evolution time over circuits derived from the traditional Quadratic Unconstrained Binary Optimization (QUBO) formalism. We illustrate our findings in the benchmark of the QUBO- and Lagrangian-based DAQC on the NP-complete 1D 0-1 knapsack problem.
Author: William Cruz-Santos
Publisher: Springer Nature
ISBN: 3031025199
Category : Mathematics
Languages : en
Pages : 105
Get Book Here
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: Kai Liu
Publisher:
ISBN:
Category : Adiabatic invariants
Languages : en
Pages : 104
Get Book Here
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: Ernesto G. Birgin
Publisher: SIAM
ISBN: 1611973368
Category : Mathematics
Languages : en
Pages : 222
Get Book Here
Book Description
This book focuses on Augmented Lagrangian techniques for solving practical constrained optimization problems. The authors: rigorously delineate mathematical convergence theory based on sequential optimality conditions and novel constraint qualifications; orient the book to practitioners by giving priority to results that provide insight on the practical behavior of algorithms and by providing geometrical and algorithmic interpretations of every mathematical result; and fully describe a freely available computational package for constrained optimization and illustrate its usefulness with applications.
Author: Kazufumi Ito
Publisher: SIAM
ISBN: 9780898718614
Category : Mathematics
Languages : en
Pages : 359
Get Book Here
Book Description
Lagrange multiplier theory provides a tool for the analysis of a general class of nonlinear variational problems and is the basis for developing efficient and powerful iterative methods for solving these problems. This comprehensive monograph analyzes Lagrange multiplier theory and shows its impact on the development of numerical algorithms for problems posed in a function space setting. The authors develop and analyze efficient algorithms for constrained optimization and convex optimization problems based on the augumented Lagrangian concept and cover such topics as sensitivity analysis, convex optimization, second order methods, and shape sensitivity calculus. General theory is applied to challenging problems in optimal control of partial differential equations, image analysis, mechanical contact and friction problems, and American options for the Black-Scholes model.
Author: David Nicholas Gosset
Publisher:
ISBN:
Category :
Languages : en
Pages : 143
Get Book Here
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: Guanglei Xu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
Rapid developments in experiments provide promising platforms for realising quantum computation and quantum simulation. This, in turn, opens new possibilities for developing useful quantum algorithms and explaining complex many-body physics. The advantages of quantum computation have been demonstrated in a small range of subjects, but the potential applications of quantum algorithms for solving complex classical problems are still under investigation. Deeper understanding of complex many-body systems can lead to realising quantum simulation to study systems which are inaccessible by other means.This thesis studies different topics of quantum computation and quantum simulation.The first one is improving a quantum algorithm in adiabatic quantum computing, which can be used to solve classical problems like combinatorial optimisation problems and simulated annealing. We are able to reach a new bound of time cost for the algorithm which has a potential to achieve a speed up over standard adiabatic quantum computing. The second topic is to understand the amplitude noise in optical lattices in the context of adiabatic state preparation and the thermalisation of the energy introduced to the system. We identify regimes where introducing certain type of noise in experiments would improve the final fidelity of adiabatic state preparation, and demonstrate the robustness of the state preparation to imperfect noise implementations. We also discuss the competition between heating and dephasing effects, the energy introduced by non-adiabaticity and heating, and the thermalisation of the system after an application of amplitude noise on the lattice. The third topic is to design quantum algorithms to solve classical problems of fluid dynamics. We develop a quantum algorithm based around phase estimation that can be tailored to specific fluid dynamics problems and demonstrate a quantum speed up over classical Monte Carlo methods. This generates new bridge between quantum physics and fluid dynamics engineering, can be used to estimate the potential impact of quantum computers and provides feedback on requirements for implementing quantum algorithms on quantum devices.
Author: Jiří Mikyška
Publisher: Springer Nature
ISBN: 3031360303
Category : Computers
Languages : en
Pages : 809
Get Book Here
Book Description
The five-volume set LNCS 14073-14077 constitutes the proceedings of the 23rd International Conference on Computational Science, ICCS 2023, held in Prague, Czech Republic, during July 3-5, 2023. The total of 188 full papers and 94 short papers presented in this book set were carefully reviewed and selected from 530 submissions. 54 full and 37 short papers were accepted to the main track; 134 full and 57 short papers were accepted to the workshops/thematic tracks. The theme for 2023, "Computation at the Cutting Edge of Science", highlights the role of Computational Science in assisting multidisciplinary research. This conference was a unique event focusing on recent developments in scalable scientific algorithms, advanced software tools; computational grids; advanced numerical methods; and novel application areas. These innovative novel models, algorithms, and tools drive new science through efficient application in physical systems, computational and systems biology, environmental systems, finance, and others.
Author: Shai Avidan
Publisher: Springer Nature
ISBN: 3031200500
Category : Computers
Languages : en
Pages : 820
Get Book Here
Book Description
The 39-volume set, comprising the LNCS books 13661 until 13699, constitutes the refereed proceedings of the 17th European Conference on Computer Vision, ECCV 2022, held in Tel Aviv, Israel, during October 23–27, 2022. The 1645 papers presented in these proceedings were carefully reviewed and selected from a total of 5804 submissions. The papers deal with topics such as computer vision; machine learning; deep neural networks; reinforcement learning; object recognition; image classification; image processing; object detection; semantic segmentation; human pose estimation; 3d reconstruction; stereo vision; computational photography; neural networks; image coding; image reconstruction; object recognition; motion estimation.
Author: Catherine C. McGeoch
Publisher: Springer Nature
ISBN: 3031025180
Category : Mathematics
Languages : en
Pages : 83
Get Book Here
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
