Quantum Computing A Shift From Bits To Qubits

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Quantum Computing: A Shift from Bits to Qubits

Author : Rajiv Pandey,Nidhi Srivastava,Neeraj Kumar Singh,Kanishka Tyagi
Publisher : Springer Nature
Page : 487 pages
File Size : 40,6 Mb
Release : 2023-03-29
Category : Technology & Engineering
ISBN : 9789811995309

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Quantum Computing: A Shift from Bits to Qubits by Rajiv Pandey,Nidhi Srivastava,Neeraj Kumar Singh,Kanishka Tyagi Pdf

The edited book is a consolidated handbook on quantum computing that covers quantum basic science and mathematics to advanced concepts and applications of quantum computing and quantum machine learning applied to diverse domains. The book includes dedicated chapters on introduction to quantum computing, its practical applications, the working behind quantum systems, quantum algorithms, quantum communications, and quantum cryptography. Each challenge that can be addressed with quantum technologies is further discussed from theoretical and practical perspectives. The book is divided into five parts: Part I: Scientific Theory for Quantum, Part II: Quantum Computing: Building Concepts, Part III: Quantum Algorithms- Theory & Applications, Part IV: Quantum Simulation Tools & Demonstrations, and Part V: Future Direction and Applications.

Dancing with Qubits

Author : Robert S. Sutor
Publisher : Packt Publishing Ltd
Page : 515 pages
File Size : 45,8 Mb
Release : 2019-11-28
Category : Computers
ISBN : 9781838825256

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Dancing with Qubits by Robert S. Sutor Pdf

Explore the principles and practicalities of quantum computing Key Features Discover how quantum computing works and delve into the math behind it with this quantum computing textbook Learn how it may become the most important new computer technology of the century Explore the inner workings of quantum computing technology to quickly process complex cloud data and solve problems Book Description Quantum computing is making us change the way we think about computers. Quantum bits, a.k.a. qubits, can make it possible to solve problems that would otherwise be intractable with current computing technology. Dancing with Qubits is a quantum computing textbook that starts with an overview of why quantum computing is so different from classical computing and describes several industry use cases where it can have a major impact. From there it moves on to a fuller description of classical computing and the mathematical underpinnings necessary to understand such concepts as superposition, entanglement, and interference. Next up is circuits and algorithms, both basic and more sophisticated. It then nicely moves on to provide a survey of the physics and engineering ideas behind how quantum computing hardware is built. Finally, the book looks to the future and gives you guidance on understanding how further developments will affect you. Really understanding quantum computing requires a lot of math, and this book doesn't shy away from the necessary math concepts you'll need. Each topic is introduced and explained thoroughly, in clear English with helpful examples. What you will learn See how quantum computing works, delve into the math behind it, what makes it different, and why it is so powerful with this quantum computing textbook Discover the complex, mind-bending mechanics that underpin quantum systems Understand the necessary concepts behind classical and quantum computing Refresh and extend your grasp of essential mathematics, computing, and quantum theory Explore the main applications of quantum computing to the fields of scientific computing, AI, and elsewhere Examine a detailed overview of qubits, quantum circuits, and quantum algorithm Who this book is for Dancing with Qubits is a quantum computing textbook for those who want to deeply explore the inner workings of quantum computing. This entails some sophisticated mathematical exposition and is therefore best suited for those with a healthy interest in mathematics, physics, engineering, and computer science.

Quantum Computing

Author : Kuldeep Singh Kaswan,Jagjit Singh Dhatterwal,Anupam Baliyan,Shalli Rani
Publisher : John Wiley & Sons
Page : 340 pages
File Size : 53,5 Mb
Release : 2023-07-13
Category : Computers
ISBN : 9781394157839

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Quantum Computing by Kuldeep Singh Kaswan,Jagjit Singh Dhatterwal,Anupam Baliyan,Shalli Rani Pdf

QUANTUM COMPUTING A helpful introduction to all aspects of quantum computing Quantum computing is a field combining quantum mechanics—the physical science of nature at the scale of atoms and subatomic particles—and information science. Where ordinary computing uses bits, logical values whose position can either be 0 or 1, quantum computing is built around qubits, a fundamental unit of quantum information which can exist in a superposition of both states. As quantum computers are able to complete certain kinds of functions more accurately and efficiently than computers built on classical binary logic, quantum computing is an emerging frontier which promises to revolutionize information science and its applications. This book provides a concise, accessible introduction to quantum computing. It begins by introducing the essentials of quantum mechanics that information and computer scientists require, before moving to detailed discussions of quantum computing in theory and practice. As quantum computing becomes an ever-greater part of the global information technology landscape, the knowledge in Quantum Computing will position readers to join a vital and highly marketable field of research and development. The book’s readers will also find: Detailed diagrams and illustrations throughout A broadly applicable quantum algorithm that improves on the best-known classical algorithms for a wide range of problems In-depth discussion of essential topics including key distribution, cluster state quantum computing, superconducting qubits, and more Quantum Computing is perfect for advanced undergraduate and graduate students in computer science, engineering, mathematics, or the physical sciences, as well as for researchers and academics at the intersection of these fields who want a concise reference.

Quantum Computing

Author : National Academies of Sciences, Engineering, and Medicine,Division on Engineering and Physical Sciences,Intelligence Community Studies Board,Computer Science and Telecommunications Board,Committee on Technical Assessment of the Feasibility and Implications of Quantum Computing
Publisher : National Academies Press
Page : 273 pages
File Size : 54,7 Mb
Release : 2019-04-27
Category : Computers
ISBN : 9780309479691

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Quantum Computing by National Academies of Sciences, Engineering, and Medicine,Division on Engineering and Physical Sciences,Intelligence Community Studies Board,Computer Science and Telecommunications Board,Committee on Technical Assessment of the Feasibility and Implications of Quantum Computing Pdf

Quantum mechanics, the subfield of physics that describes the behavior of very small (quantum) particles, provides the basis for a new paradigm of computing. First proposed in the 1980s as a way to improve computational modeling of quantum systems, the field of quantum computing has recently garnered significant attention due to progress in building small-scale devices. However, significant technical advances will be required before a large-scale, practical quantum computer can be achieved. Quantum Computing: Progress and Prospects provides an introduction to the field, including the unique characteristics and constraints of the technology, and assesses the feasibility and implications of creating a functional quantum computer capable of addressing real-world problems. This report considers hardware and software requirements, quantum algorithms, drivers of advances in quantum computing and quantum devices, benchmarks associated with relevant use cases, the time and resources required, and how to assess the probability of success.

Quantum Computing for the Quantum Curious

Author : Ciaran Hughes,Joshua Isaacson,Anastasia Perry,Ranbel F. Sun,Jessica Turner
Publisher : Springer Nature
Page : 159 pages
File Size : 45,8 Mb
Release : 2021-03-22
Category : Science
ISBN : 9783030616014

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Quantum Computing for the Quantum Curious by Ciaran Hughes,Joshua Isaacson,Anastasia Perry,Ranbel F. Sun,Jessica Turner Pdf

This open access book makes quantum computing more accessible than ever before. A fast-growing field at the intersection of physics and computer science, quantum computing promises to have revolutionary capabilities far surpassing “classical” computation. Getting a grip on the science behind the hype can be tough: at its heart lies quantum mechanics, whose enigmatic concepts can be imposing for the novice. This classroom-tested textbook uses simple language, minimal math, and plenty of examples to explain the three key principles behind quantum computers: superposition, quantum measurement, and entanglement. It then goes on to explain how this quantum world opens up a whole new paradigm of computing. The book bridges the gap between popular science articles and advanced textbooks by making key ideas accessible with just high school physics as a prerequisite. Each unit is broken down into sections labelled by difficulty level, allowing the course to be tailored to the student’s experience of math and abstract reasoning. Problem sets and simulation-based labs of various levels reinforce the concepts described in the text and give the reader hands-on experience running quantum programs. This book can thus be used at the high school level after the AP or IB exams, in an extracurricular club, or as an independent project resource to give students a taste of what quantum computing is really about. At the college level, it can be used as a supplementary text to enhance a variety of courses in science and computing, or as a self-study guide for students who want to get ahead. Additionally, readers in business, finance, or industry will find it a quick and useful primer on the science behind computing’s future.

Quantum Computing and Quantum Bits in Mesoscopic Systems

Author : Anthony Leggett,Berardo Ruggiero,Paolo Silvestrini
Publisher : Springer Science & Business Media
Page : 264 pages
File Size : 43,6 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781441990921

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Quantum Computing and Quantum Bits in Mesoscopic Systems by Anthony Leggett,Berardo Ruggiero,Paolo Silvestrini Pdf

Quantum information science is a new field of science and technology which requires the collaboration of researchers coming from different fields of physics, mathematics, and engineering: both theoretical and applied. Quantum Computing and Quantum Bits in Mesoscopic Systems addresses fundamental aspects of quantum physics, enhancing the connection between the quantum behavior of macroscopic systems and information theory. In addition to theoretical quantum physics, the book comprehensively explores practical implementation of quantum computing and information processing devices. On the experimental side, this book reports on recent and previous observations of quantum behavior in several physical systems, coherently coupled Bose-Einstein condensates, quantum dots, superconducting quantum interference devices, Cooper pair boxes, and electron pumps in the context of the Josephson effect. In these systems, the book discusses all required steps, from fabrication through characterization to the final basic implementation for quantum computing.

Quantum Computing from the Ground Up

Author : Riley Tipton Perry
Publisher : World Scientific Publishing Company
Page : 256 pages
File Size : 47,6 Mb
Release : 2012-07-11
Category : Science
ISBN : 9789814412131

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Quantum Computing from the Ground Up by Riley Tipton Perry Pdf

Quantum computing — the application of quantum mechanics to information — represents a fundamental break from classical information and promises to dramatically increase a computer's power. Many difficult problems, such as the factorization of large numbers, have so far resisted attack by classical computers yet are easily solved with quantum computers. If they become feasible, quantum computers will end standard practices such as RSA encryption. Most of the books or papers on quantum computing require (or assume) prior knowledge of certain areas such as linear algebra or quantum mechanics. The majority of the currently-available literature is hard to understand for the average computer enthusiast or interested layman. This text attempts to teach quantum computing from the ground up in an easily readable way, providing a comprehensive tutorial that includes all the necessary mathematics, computer science and physics. Errata(s) Errata

Quantum Computing

Author : Eleanor G. Rieffel,Wolfgang H. Polak
Publisher : MIT Press
Page : 389 pages
File Size : 41,7 Mb
Release : 2014-08-29
Category : Computers
ISBN : 9780262526678

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Quantum Computing by Eleanor G. Rieffel,Wolfgang H. Polak Pdf

A thorough exposition of quantum computing and the underlying concepts of quantum physics, with explanations of the relevant mathematics and numerous examples. The combination of two of the twentieth century's most influential and revolutionary scientific theories, information theory and quantum mechanics, gave rise to a radically new view of computing and information. Quantum information processing explores the implications of using quantum mechanics instead of classical mechanics to model information and its processing. Quantum computing is not about changing the physical substrate on which computation is done from classical to quantum but about changing the notion of computation itself, at the most basic level. The fundamental unit of computation is no longer the bit but the quantum bit or qubit. This comprehensive introduction to the field offers a thorough exposition of quantum computing and the underlying concepts of quantum physics, explaining all the relevant mathematics and offering numerous examples. With its careful development of concepts and thorough explanations, the book makes quantum computing accessible to students and professionals in mathematics, computer science, and engineering. A reader with no prior knowledge of quantum physics (but with sufficient knowledge of linear algebra) will be able to gain a fluent understanding by working through the book.

Dancing with Qubits

Author : Robert S. Sutor
Publisher : Packt Publishing Ltd
Page : 685 pages
File Size : 46,7 Mb
Release : 2024-03-28
Category : Computers
ISBN : 9781837634620

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Dancing with Qubits by Robert S. Sutor Pdf

Unlock the core math and understand the technical nuances of quantum computing in this detailed guide. Delve into the practicality of NISQ algorithms, and survey promising advancements in quantum machine learning. Key Features Discover how quantum computing works and delve into the math behind it with practical examples Learn about and assess the most up-to-date quantum computing topics including quantum machine learning Explore the inner workings of existing quantum computing technologies to understand how they may perform significantly better than their classical counterparts Book DescriptionDancing with Qubits, Second Edition, is a comprehensive quantum computing textbook that starts with an overview of why quantum computing is so different from classical computing and describes several industry use cases where it can have a major impact. A full description of classical computing and the mathematical underpinnings of quantum computing follows, helping you better understand concepts such as superposition, entanglement, and interference. Next up are circuits and algorithms, both basic and sophisticated, as well as a survey of the physics and engineering ideas behind how quantum computing hardware is built. Finally, the book looks to the future and gives you guidance on understanding how further developments may affect you. This new edition is updated throughout with more than 100 new exercises and includes new chapters on NISQ algorithms and quantum machine learning. Understanding quantum computing requires a lot of math, and this book doesn't shy away from the necessary math concepts you'll need. Each topic is explained thoroughly and with helpful examples, leaving you with a solid foundation of knowledge in quantum computing that will help you pursue and leverage quantum-led technologies.What you will learn Explore the mathematical foundations of quantum computing Discover the complex, mind-bending concepts that underpin quantum systems Understand the key ideas behind classical and quantum computing Refresh and extend your grasp of essential mathematics, computing, and quantum theory Examine a detailed overview of qubits and quantum circuits Dive into quantum algorithms such as Grover’s search, Deutsch-Jozsa, Simon’s, and Shor’s Explore the main applications of quantum computing in the fields of scientific computing, AI, and elsewhere Who this book is for Dancing with Qubits, Second Edition, is a quantum computing textbook for all those who want to understand and explore the inner workings of quantum computing. This entails building up from basic to some sophisticated mathematics and is therefore best suited for those with a healthy interest in mathematics, physics, engineering, or computer science.

Explorations in Quantum Computing

Author : Colin P. Williams
Publisher : Springer Science & Business Media
Page : 724 pages
File Size : 40,6 Mb
Release : 2010-12-07
Category : Computers
ISBN : 9781846288876

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Explorations in Quantum Computing by Colin P. Williams Pdf

By the year 2020, the basic memory components of a computer will be the size of individual atoms. At such scales, the current theory of computation will become invalid. "Quantum computing" is reinventing the foundations of computer science and information theory in a way that is consistent with quantum physics - the most accurate model of reality currently known. Remarkably, this theory predicts that quantum computers can perform certain tasks breathtakingly faster than classical computers – and, better yet, can accomplish mind-boggling feats such as teleporting information, breaking supposedly "unbreakable" codes, generating true random numbers, and communicating with messages that betray the presence of eavesdropping. This widely anticipated second edition of Explorations in Quantum Computing explains these burgeoning developments in simple terms, and describes the key technological hurdles that must be overcome to make quantum computers a reality. This easy-to-read, time-tested, and comprehensive textbook provides a fresh perspective on the capabilities of quantum computers, and supplies readers with the tools necessary to make their own foray into this exciting field. Topics and features: concludes each chapter with exercises and a summary of the material covered; provides an introduction to the basic mathematical formalism of quantum computing, and the quantum effects that can be harnessed for non-classical computation; discusses the concepts of quantum gates, entangling power, quantum circuits, quantum Fourier, wavelet, and cosine transforms, and quantum universality, computability, and complexity; examines the potential applications of quantum computers in areas such as search, code-breaking, solving NP-Complete problems, quantum simulation, quantum chemistry, and mathematics; investigates the uses of quantum information, including quantum teleportation, superdense coding, quantum data compression, quantum cloning, quantum negation, and quantum cryptography; reviews the advancements made towards practical quantum computers, covering developments in quantum error correction and avoidance, and alternative models of quantum computation. This text/reference is ideal for anyone wishing to learn more about this incredible, perhaps "ultimate," computer revolution. Dr. Colin P. Williams is Program Manager for Advanced Computing Paradigms at the NASA Jet Propulsion Laboratory, California Institute of Technology, and CEO of Xtreme Energetics, Inc. an advanced solar energy company. Dr. Williams has taught quantum computing and quantum information theory as an acting Associate Professor of Computer Science at Stanford University. He has spent over a decade inspiring and leading high technology teams and building business relationships with and Silicon Valley companies. Today his interests include terrestrial and Space-based power generation, quantum computing, cognitive computing, computational material design, visualization, artificial intelligence, evolutionary computing, and remote olfaction. He was formerly a Research Scientist at Xerox PARC and a Research Assistant to Prof. Stephen W. Hawking, Cambridge University.

Quantum Computer Science

Author : Marco Lanzagorta,Jeffrey K. Uhlmann
Publisher : Morgan & Claypool Publishers
Page : 125 pages
File Size : 48,6 Mb
Release : 2009
Category : Algorithms
ISBN : 9781598297324

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Quantum Computer Science by Marco Lanzagorta,Jeffrey K. Uhlmann Pdf

In this text we present a technical overview of the emerging field of quantum computation along with new research results by the authors. What distinguishes our presentation from that of others is our focus on the relationship between quantum computation and computer science. Specifically, our emphasis is on the computational model of quantum computing rather than on the engineering issues associated with its physical implementation. We adopt this approach for the same reason that a book on computer programming doesn't cover the theory and physical realization of semiconductors. Another distinguishing feature of this text is our detailed discussion of the circuit complexity of quantum algorithms. To the extent possible we have presented the material in a form that is accessible to the computer scientist, but in many cases we retain the conventional physics notation so that the reader will also be able to consult the relevant quantum computing literature. Although we expect the reader to have a solid understanding of linear algebra, we do not assume a background in physics. This text is based on lectures given as short courses and invited presentations around the world, and it has been used as the primary text for a graduate course at George Mason University. In all these cases our challenge has been the same: how to present to a general audience a concise introduction to the algorithmic structure and applications of quantum computing on an extremely short period of time. The feedback from these courses and presentations has greatly aided in making our exposition of challenging concepts more accessible to a general audience. Table of Contents: Introduction / The Algorithmic Structure of Quantum Computing / Advantages and Limitations of Quantum Computing / Amplitude Amplification / Case Study: Computational Geometry / The Quantum Fourier Transform / Case Study: The Hidden Subgroup / Circuit Complexity Analysis of Quantum Algorithms / Conclusions / Bibliography

Quantum Computing in Action

Author : Johan Vos
Publisher : Simon and Schuster
Page : 264 pages
File Size : 43,9 Mb
Release : 2022-03-22
Category : Computers
ISBN : 9781638350941

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Quantum Computing in Action by Johan Vos Pdf

Quantum computing is on the horizon and you can get started today! This practical, clear-spoken guide shows you don’t need a physics degree to write your first quantum software. In Quantum Computing in Action you will learn: An introduction to the core concepts of quantum computing Qubits and quantum gates Superposition, entanglement, and hybrid computing Quantum algorithms including Shor’s, Deutsch-jozsa, and Grover’s search Quantum Computing in Action shows you how to leverage your existing Java skills into writing your first quantum software, so you’re ready for the quantum revolution. This book is focused on practical implementations of quantum computing algorithms—there’s no deep math or confusing theory. Using Strange, a Java-based quantum computer simulator, you’ll go hands-on with quantum computing’s core components including qubits and quantum gates. About the technology Quantum computing promises unimaginably fast performance for tasks like encryption, scientific modeling, manufacturing logistics, financial modeling, and AI. Developers can explore quantum computing now using free simulators, and increasingly powerful true quantum systems are gradually becoming available for production use. This book gives you a head start on quantum computing by introducing core concepts, key algorithms, and the most beneficial use cases. About the book Quantum Computing in Action is a gentle introduction to the ideas and applications of quantum computing. After briefly reviewing the science that makes quantum tick, it guides you through practical implementations of quantum computing algorithms. You’ll write your first quantum code and explore qubits and quantum gates with the Java-based Strange quantum simulator. You’ll enjoy the interesting examples and insightful explanations as you create quantum algorithms using standard Java and your favorite IDE and build tools. What's inside An introduction to the core concepts of quantum computing Qubits and quantum gates Superposition, entanglement, and hybrid computing Quantum algorithms including Shor’s, Deutsch-jozsa, and Grover’s search About the reader For Java developers. No advanced math knowledge required. About the author Johan Vos is a cofounder of Gluon, a Java technology company. He is a Java Champion and holds an MSc in Mining Engineering and a PhD in Applied Physics. Table of Contents PART 1 QUANTUM COMPUTING INTRODUCTION 1 Evolution, revolution, or hype? 2 “Hello World,” quantum computing style 3 Qubits and quantum gates: The basic units in quantum computing PART 2 FUNDAMENTAL CONCEPTS AND HOW THEY RELATE TO CODE 4 Superposition 5 Entanglement 6 Quantum networking: The basics PART 3 QUANTUM ALGORITHMS AND CODE 7 Our HelloWorld, explained 8 Secure communication using quantum computing 9 Deutsch-Jozsa algorithm 10 Grover’s search algorithm 11 Shor’s algorithm

Quantum Computing in Solid State Systems

Author : Berardo Ruggiero,Per Delsing,Carmine Granata,Yuri A. Pashkin,P. Silvestrini
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 50,5 Mb
Release : 2006-05-30
Category : Science
ISBN : 9780387311432

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Quantum Computing in Solid State Systems by Berardo Ruggiero,Per Delsing,Carmine Granata,Yuri A. Pashkin,P. Silvestrini Pdf

Quantum Computation in Solid State Systems discusses experimental implementation of quantum computing for information processing devices; in particular observations of quantum behavior in several solid state systems are presented. The complementary theoretical contributions provide models of minimizing decoherence in the different systems. Most recent theoretical and experimental results on macroscopic quantum coherence of mesoscopic systems, as well as the realization of solid-state qubits and quantum gates are discussed. Particular attention is given to coherence effects in Josephson devices. Other solid state systems---including quantum dots, optical, ion, and spin devices---are also discussed.

The Revolution in Quantum Technology

Author : Emil Kowalski
Publisher : Unknown
Page : 0 pages
File Size : 50,5 Mb
Release : 2023-12-31
Category : Computers
ISBN : 9360927465

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The Revolution in Quantum Technology by Emil Kowalski Pdf

If you want to learn more about cutting edge quantum technology, read "The Revolution in Quantum Technology: How Game-Changing Tools Are Altering Our World." The author of this new book, Emil Kowalski, looks at how quantum technology will change our quickly changing digital world. Find out what quantum computing is all about. In this type of computing, quantum bits (qubits) are used instead of regular bits. This lets many computers work together to solve hard problems very quickly. Learn about the power of quantum communication, which changes how we share private information in a world that is becoming more and more connected by creating safe, unhackable lines of communication. Kowalski goes into detail about the rules of quantum mechanics, which makes the difference between conventional and quantum computing very clear. As the digital world goes through a paradigm shift, it's important to understand these key ideas in order to find your way around the quantum world. Learn more about the interesting world of quantum bits (qubits) and superposition, where quantum objects can be in more than one state at the same time. Kowalski takes the mystery out of the idea of superposition and the different ways it could be used. This helps the reader understand how complicated quantum systems work. Take a trip into the world of entanglement, where quantum objects become linked and can change each other's states even when they are far apart. Kowalski explains the part that entanglement plays in quantum computing, which opens up new ways to communicate and do computations. Learn about quantum gates and circuits, which are the building blocks of quantum programs. The author explains how these quantum structures will lead to huge improvements in computer power and the ability to solve problems. As the story goes on, Kowalski talks about quantum algorithms and how they might affect many areas, from security to solving optimization problems. Look at how we're about to change the way we solve hard problems and look at data in the quantum age. "The Revolution in Quantum Technology" also explains the wide range of uses that can be improved by quantum technology, such as safe communication networks and better imaging methods. Find out how quantum technology is going to change fields like business, healthcare, and telecommunications. In a world where privacy and security are very important, this book looks at the flaws in traditional cryptography and presents the idea of quantum key distribution (QKD). Learn how quantum encryption, which can't be broken, makes sure that information is sent safely in a time when online threats are growing. This important and insightful book is a must-read for anyone who wants to understand how quantum technology could change the world. Learn all about how these game-changing tools are changing our world by diving into the quantum revolution with Emil Kowalski.

An Introduction to Quantum Computing Algorithms

Author : Arthur O. Pittenger
Publisher : Springer Science & Business Media
Page : 149 pages
File Size : 46,7 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9781461213901

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An Introduction to Quantum Computing Algorithms by Arthur O. Pittenger Pdf

In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com puter. Since the difficulty of the factoring problem is crucial for the se curity of a public key encryption system, interest (and funding) in quan tum computing and quantum computation suddenly blossomed. Quan tum computing had arrived. The study of the role of quantum mechanics in the theory of computa tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system.