Download or read book Quantum Algorithms for Searching Resampling and Hidden Shift Problems written by Māris Ozols and published by . This book was released on 2012 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is on quantum algorithms. It has three main themes: (1) quantum walk based search algorithms, (2) quantum rejection sampling, and (3) the Boolean function hidden shift problem. The first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. In the first part of this thesis we show how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster. More formally, given a random walk on a graph with an unknown set of marked vertices, we construct a quantum walk that finds a marked vertex in a number of steps that is quadratically smaller than the hitting time of the random walk. The main idea of our approach is to interpolate the random walk from one that does not stop when a marked vertex is found to one that stops. The quantum equivalent of this procedure drives the initial superposition over all vertices to a superposition over marked vertices. We present an adiabatic as well as a circuit version of our algorithm, and apply it to the spatial search problem on the 2D grid. In the second part we study a quantum version of the problem of resampling one probability distribution to another. More formally, given query access to a black box that produces a coherent superposition of unknown quantum states with given amplitudes, the problem is to prepare a coherent superposition of the same states with different specified amplitudes. Our main result is a tight characterization of the number of queries needed for this transformation. By utilizing the symmetries of the problem, we prove a lower bound using a hybrid argument and semidefinite programming. For the matching upper bound we construct a quantum algorithm that generalizes the rejection sampling method first formalized by von~Neumann in~1951. We describe quantum algorithms for the linear equations problem and quantum Metropolis sampling as applications of quantum rejection sampling. In the third part we consider a hidden shift problem for Boolean functions: given oracle access to f(x+s), where f(x) is a known Boolean function, determine the hidden shift s. We construct quantum algorithms for this problem using the "pretty good measurement" and quantum rejection sampling. Both algorithms use the Fourier transform and their complexity can be expressed in terms of the Fourier spectrum of f (in particular, in the second case it relates to "water-filling" of the spectrum). We also construct algorithms for variations of this problem where the task is to verify a given shift or extract only a single bit of information about it.

Download or read book Search for New Quantum Algorithms written by and published by . This book was released on 2006 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first objective of this effort, searching for new quantum algorithms, created six new quantum hidden subgroup algorithms. The second objective, improving the theoretical understanding of existing quantum algorithms, produced three new systematic procedures. Also, application of combinatorial group theory led to substantial progress in the understanding and analysis of nonabelian quantum hidden subgroup algorithms. Additionally, methods and techniques of quantum topology have been used to obtain new results in quantum computing including discovery of a relationship between quantum entanglement and topological linking. The last objective, analyzing issues associated with algorithm implementation proposed distributed quantum computing (DQC) as a fast track to scalable quantum computing with technology available within the next five years. A universal set of DQC primitives has been created and used to transform the quantum Fourier transform and the Shor algorithm into DQC. The additional computational overhead needed for DQC algorithms is insignificant and DQC is found to simplify the decoherence problem.

Download or read book Quantum Algorithms and the Fourier Transform written by Lawrence Poi Heng Ip and published by . This book was released on 2004 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Continuous time Quantum Algorithms electronic Resource Searching and Adiabatic Computation written by Lawrence Mario Ioannou and published by University of Waterloo. This book was released on 2002 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e.g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. Soon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi et. al. [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. I begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework.

Download or read book Quantum Walks and Search Algorithms written by Renato Portugal and published by Springer. This book was released on 2015-02-08 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses an interesting area of quantum computation called quantum walks, which play an important role in building quantum algorithms, in particular search algorithms. Quantum walks are the quantum analogue of classical random walks. It is known that quantum computers have great power for searching unsorted databases. This power extends to many kinds of searches, particularly to the problem of finding a specific location in a spatial layout, which can be modeled by a graph. The goal is to find a specific node knowing that the particle uses the edges to jump from one node to the next. This book is self-contained with main topics that include: Grover's algorithm, describing its geometrical interpretation and evolution by means of the spectral decomposition of the evolution operator Analytical solutions of quantum walks on important graphs like line, cycles, two-dimensional lattices, and hypercubes using Fourier transforms Quantum walks on generic graphs, describing methods to calculate the limiting distribution and mixing time Spatial search algorithms, with emphasis on the abstract search algorithm (the two-dimensional lattice is used as an example) Szedgedy's quantum-walk model and a natural definition of quantum hitting time (the complete graph is used as an example) The reader will benefit from the pedagogical aspects of the book, learning faster and with more ease than would be possible from the primary research literature. Exercises and references further deepen the reader's understanding, and guidelines for the use of computer programs to simulate the evolution of quantum walks are also provided.

Download or read book Examining Quantum Algorithms for Quantum Image Processing written by Haisheng Li and published by Engineering Science Reference. This book was released on 2020 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book provides a comprehensive introduction to quantum image algorithms to establish frameworks of quantum image processing"--

Download or read book On Quantum Algorithms for Noncommutative Hidden Subgroups written by and published by . This book was released on 1998 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum algorithms for factoring and discrete logarithm have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. The authors present a quantum algorithm for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function. They also explore the difficulties of developing an algorithm to process the data to explicitly calculate a generating set for the subgroup. A general framework for the noncommutative hidden subgroup problem is discussed and they indicate future research directions.

Download or read book Statistical Performance of Ensemble Quantum Computers Applied to Search Problems written by Tomasz M. Kott and published by . This book was released on 2006 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: The statistical performance of quantum search algorithms is important for Nuclear Magnetic Resonance (NMR) ensemble quantum computing. To compare the performance of a quantum algorithm on an ensemble quantum computer, we compare it to classical probabilistic algorithms.

Download or read book Quantum Algorithms for a Few Problems written by Siddharth S. Raval and published by . This book was released on 2011 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Quantum Algorithms for written by Hamed Ahmadi and published by . This book was released on 2012 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of black-box modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT)) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In the second part of this dissertation, we introduce an alternative approach to approximately implement QPE with arbitrary constant-precision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev's original approach. For approximating the eigenphase precise to the nth bit, Kitaev's original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev's approach. The other problem we investigate relates to approximating the Tutte polynomial. We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q,1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at a point is (e[superscripts 2pi i/5], e[superscripts -2pi i/5]) is DQC1-complete and at some points (q[superscript k], 1 + [1-q[superscript -k]/(q[superscript 1/2]-q[superscript -1/2])[superscript 2]) for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones-Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid bsuch that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of the resulting braid [tilde over b] are alternating links. The final part of the dissertation focuses on finding the structure of a black-box module or algebra. Suppose we are given black-box access to a finite module M or algebra over a finite ring R and a list of generators for M and R. We show how to find a linear basis and structure constants for M in quantum poly (log [vertical line]M[vertical line]) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for rings. We then show that our algorithm is a useful primitive allowing quantum computer to determine the structure of a finite associative algebra as a direct sum of simple algebras. Moreover, it solves a wide variety of problems regarding finite modules and rings. Although our quantum algorithm is based on Abelian Fourier transforms, it solves problems regarding the multiplicative structure of modules and algebras, which need not be commutative. Examples include finding the intersection and quotient of two modules, finding the additive and multiplicative identities in a module, computing the order of an module, solving linear equations over modules, deciding whether an ideal is maximal, finding annihilators, and testing the injectivity and surjectivity of ring homomorphisms. These problems appear to be exponentially hard classically.

Download or read book Supervised Learning with Quantum Computers written by Maria Schuld and published by Springer. This book was released on 2018-08-30 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum machine learning investigates how quantum computers can be used for data-driven prediction and decision making. The books summarises and conceptualises ideas of this relatively young discipline for an audience of computer scientists and physicists from a graduate level upwards. It aims at providing a starting point for those new to the field, showcasing a toy example of a quantum machine learning algorithm and providing a detailed introduction of the two parent disciplines. For more advanced readers, the book discusses topics such as data encoding into quantum states, quantum algorithms and routines for inference and optimisation, as well as the construction and analysis of genuine ``quantum learning models''. A special focus lies on supervised learning, and applications for near-term quantum devices.

Download or read book Exact Algorithms for Constraint Satisfaction Problems written by Robin Alexander Moser and published by Logos Verlag Berlin GmbH. This book was released on 2013 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Boolean satisfiability problem (SAT) and its generalization to variables of higher arities - constraint satisfaction problems (CSP) - can arguably be called the most "natural" of all NP-complete problems. The present work is concerned with their algorithmic treatment. It consists of two parts. The first part investigates CSPs for which satisfiability follows from the famous Lovasz Local Lemma. Since its discovery in 1975 by Paul Erdos and Laszlo Lovasz, it has been known that CSPs without dense spots of interdependent constraints always admit a satisfying assignment. However, an iterative procedure to discover such an assignment was not available. We refine earlier attempts at making the Local Lemma algorithmic and present a polynomial time algorithm which is able to make almost all known applications constructive. In the second part, we leave behind the class of polynomial time tractable problems and instead investigate the randomized exponential time algorithm devised and analyzed by Uwe Schoning in 1999, which solves arbitrary clause satisfaction problems. Besides some new interesting perspectives on the algorithm, the main contribution of this part consists of a refinement of earlier approaches at derandomizing Schoning's algorithm. We present a deterministic variant which losslessly reaches the performance of the randomized original.

Download or read book Machine Learning Meets Quantum Physics written by Kristof T. Schütt and published by Springer Nature. This book was released on 2020-06-03 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: Designing molecules and materials with desired properties is an important prerequisite for advancing technology in our modern societies. This requires both the ability to calculate accurate microscopic properties, such as energies, forces and electrostatic multipoles of specific configurations, as well as efficient sampling of potential energy surfaces to obtain corresponding macroscopic properties. Tools that can provide this are accurate first-principles calculations rooted in quantum mechanics, and statistical mechanics, respectively. Unfortunately, they come at a high computational cost that prohibits calculations for large systems and long time-scales, thus presenting a severe bottleneck both for searching the vast chemical compound space and the stupendously many dynamical configurations that a molecule can assume. To overcome this challenge, recently there have been increased efforts to accelerate quantum simulations with machine learning (ML). This emerging interdisciplinary community encompasses chemists, material scientists, physicists, mathematicians and computer scientists, joining forces to contribute to the exciting hot topic of progressing machine learning and AI for molecules and materials. The book that has emerged from a series of workshops provides a snapshot of this rapidly developing field. It contains tutorial material explaining the relevant foundations needed in chemistry, physics as well as machine learning to give an easy starting point for interested readers. In addition, a number of research papers defining the current state-of-the-art are included. The book has five parts (Fundamentals, Incorporating Prior Knowledge, Deep Learning of Atomistic Representations, Atomistic Simulations and Discovery and Design), each prefaced by editorial commentary that puts the respective parts into a broader scientific context.

Download or read book Machine Learning with Quantum Computers written by Maria Schuld and published by Springer Nature. This book was released on 2021-10-17 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an introduction into quantum machine learning research, covering approaches that range from "near-term" to fault-tolerant quantum machine learning algorithms, and from theoretical to practical techniques that help us understand how quantum computers can learn from data. Among the topics discussed are parameterized quantum circuits, hybrid optimization, data encoding, quantum feature maps and kernel methods, quantum learning theory, as well as quantum neural networks. The book aims at an audience of computer scientists and physicists at the graduate level onwards. The second edition extends the material beyond supervised learning and puts a special focus on the developments in near-term quantum machine learning seen over the past few years.

Download or read book Proceedings of Sixth International Congress on Information and Communication Technology written by Xin-She Yang and published by Springer Nature. This book was released on 2021-09-23 with total page 982 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers selected high-quality research papers presented at the Sixth International Congress on Information and Communication Technology, held at Brunel University, London, on February 25–26, 2021. It discusses emerging topics pertaining to information and communication technology (ICT) for managerial applications, e-governance, e-agriculture, e-education and computing technologies, the Internet of things (IoT) and e-mining. Written by respected experts and researchers working on ICT, the book offers a valuable asset for young researchers involved in advanced studies. The book is presented in four volumes.

Download or read book Mathematics of Public Key Cryptography written by Steven D. Galbraith and published by Cambridge University Press. This book was released on 2012-03-15 with total page 631 pages. Available in PDF, EPUB and Kindle. Book excerpt: This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography.

Download or read book Pattern Recognition and Machine Learning written by Christopher M. Bishop and published by Springer. This book was released on 2016-08-23 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first textbook on pattern recognition to present the Bayesian viewpoint. The book presents approximate inference algorithms that permit fast approximate answers in situations where exact answers are not feasible. It uses graphical models to describe probability distributions when no other books apply graphical models to machine learning. No previous knowledge of pattern recognition or machine learning concepts is assumed. Familiarity with multivariate calculus and basic linear algebra is required, and some experience in the use of probabilities would be helpful though not essential as the book includes a self-contained introduction to basic probability theory.