A topological approach to quantum mechanics and the philosophy of nature. Lectures on topological codes and quantum computation. Topological quantum computation microsoft research. It is here that the proposals for topological quantum computing seem promising4,10,11.
This iceberg is a model of quantum computation based on topological, rather than local, degrees of freedom. March 27, 2018 acknowledgments in this book i tried to present some of the knowledge and understanding i acquired in my four decades in the eld. Xiaogang wen is a theoretical condensed matter physicist, recognized for his work on introducing the notion topological order 1989 and developing the theories of this new class of quantum states of matter. Introduction to topological quantum computation university of leeds. The advantage of a quantum computer based on quantum braids. Quantum information meets quantum matter from quantum. Apr 12, 2012 in this book, a variety of different topics are presented together for the first time, forming a thorough introduction to topological quantum computation. Geometric algebraic and topological methods for quantum field theory pdf geometric algebraic and topological methods for quantum field theory pdf. Topological quantum computation is an approach to storing and manipulating quantum infor mation that. This pedagogical introduction to topological quantum computation includes the following parts. Information is something that can be encoded in the state of a physical system, and a computation is a task that can be performed with a physically realizable device. Kitaev, faulttolerant quantum computation by anyons. The makings of topological systems, their properties and their computational power are presented in a pedagogical way. H e y incompressible electron liquids with conductance quantized to an extreme precision at rational values of the o.
Pachos zhenghan wang 1 quantum information processing volume 11, pages 1969 1973 2012 cite this article. In these notes, we will learn that quantum computers di er from classical computation in several respects when it comes to the processing of the information. The braiding and fusion of anyonic excitations in quantum hall electron liquids and 2dmagnets are modeled by modular functors, opening a new. From the generalization of the topological basis for the two. The book covers everything required to understand topological faulttolerant quantum computation, ran.
Geometric algebraic and topological methods for quantum. Clearly states what is needed to achieve such huge task. The main source of this knowledge was the theory of computation community, which has been my academic and social home throughout this period. Topological systems can serve as quantum memories or as quantum computers. This thesis deals with topological orders from two different perspectives. Email your librarian or administrator to recommend adding this book to your organisations collection. Characterized by bulk dirac or weyl cones and surface fermiarc states, topological semimetals have sparked enormous research interest in recent years. They underlie the jones polynomial and arise in wittenchernsimons theory. In particular we consider the properties of anyons and their relation to topological quantum computation.
The firm has been developing topological quantum computing for more than a decade and today has researchers writing software for future machines, and working with academic laboratories to. The freedmankitaev topological model for quantum computation is an inherently faulttolerant computation scheme, storing information in topological rather than local degrees of freedom with quantum gates typically realized by braiding quasiparticles in two dimensional media. In mathematical terms, these are unitary topological modular functors. The book by pachos can be viewed as an extended version of the present. May 29, 2010 topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. Majorana zero modes and topological quantum computation npj. Topological orders with spins and fermions quantum.
Hansson anyon school berlin, 20 the quantum hall liquids t. This book expands the plan of the authors 2008 nsfcbms lectures on knots and topological quantum computing, and is intended as a primer for mathematically. Quantum computation with topological codes from qubit to. In this book, a variety of different topics are presented together for the first time, forming a thorough introduction to topological quantum computation. Hopefully i will footnote place where important things are swept under the rug. An important objective is to nd quantum algorithms that are signi cantly faster than any classical algorithm solving the same problem. The basic concepts and useful tools underlying faulttolerant quantum computation, such as universal quantum computation, stabilizer formalism, and measurementbased quantum computation, are also provided in a pedagogical way. The number of physical qubits needed by the quantum hardware and the amount of time required to implement an algorithm is dictated by the manner in which this universal quantum state is consumed.
In chapter 2, the author explains anyon statistics as geometric berry phases. We explore the mathematical foundations of topological quantum computation, a quantum computation model that is based on principles of topology which as a result is more resistant to quantum decoherence than existing models. In the case of topological quantum computing, a qubit is encoded in non local properties of a system and hence is not subject to. In this book, the author presents a variety of different topics developed together for the first time, forming an excellent introduction to topological quantum computation. Quantum computation with topological codes springerlink. Anyons are interesting on their own right in funda. As quantum technologies advance, we get closer to finding solutions to some of the worlds most challenging problems. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by braiding nonabelian anyons. Such topological insulators may provide new routes to generating novel phases and particles, possibly finding uses in technological applications in spintronics and quantum computing. Topological orders with spins and fermions springerlink. Then i keep track of uptodate results on topological quantum computing, quantum information theory, and other related subjects. A quantum network is a device consisting of quantum logic gates whose computational steps are synchronised in time.
Introduction to topological quantum computation by jiannis. Hastings, chetan nayak,1,2 jason alicea,3,4 karsten flensberg, 5stephan plugge, 6 yuval oreg, 7 charles m. Sep 21, 2012 introduction to topological quantum computation by j. American mathematical society, providence, ri 2010 download references. The power of quantum computing is based on several phenomena and laws of the quantum world that are. Provides a comprehensive introduction to topological quantum codes and faulttolerant quantum computation with them. First we provide an introduction to anyons and topological models. Introduction to topological quantum computation by j. In these states, localized particlelike excitations quasiparticles possess quantum numbers which are in many ways analogous to ordinary spin quantum numbers. Topological quantum computation using majorana fermions. His main research interest is in quantum physics and quantum information, where he has published 70 academic papers.
The book introduce a difficult concepts on topological quantum computing. Readers who are interested in the more physical aspects of topological quantum computation are recommended to read it. I will not be as precise as mathematicians usually want. That is, any of these models can efficiently simulate any of the others. Are quantum computers more powerful than classical computers. Combining physics, mathematics and computer science, topological quantum computation is a rapidly expanding research area focused on the exploration of quantum evolutions that are immune to errors. Topological quantum computation perimeter institute. Quantum topological states of matter with applications in solid state physics, quantum computing, and quantum information theory a white paper on applications of m. Read and download pdf ebook introduction to topological quantum computation 1st edition at online ebook library. Introduction to topological quantum computation 1, pachos.
Topological quantum computation ubc library open collections. Oct 27, 2015 topological quantum computation 1,2 is an approach to faulttolerant quantum computation in which the unitary quantum gates result from the braiding of certain topological quantum objects called. Kitaev, we explore the connection between faulttolerant quantum computation and nonabelian quantum statistics in two spatial dimensions. Keeping highlevel and technical language to a minimum, the author adopts a pedagogical style, making the book accessible to nonspecialists and researchers from a variety of subdisciplines.
Topological quantum computation ams bookstore american. To fully embrace the power and potential of quantum computing, the system must be engineered to meet the demands of the solutions the world needs most. Chapter 2 kau man knot invariant and relation to physics 2. Topological quantum computation zhenghan wang ucsb math. In physics, topological order is a kind of order in the zerotemperature phase of matter also known as quantum matter. Im typing as we go so please forgive all typos and unclear parts etc. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by. The computational answer is accessed by bringing anyons together and observing the result. Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. Oct 21, 2016 microsoft has been working on topological quantum computing for more than a decade, for most of which the necessary qubits were hypothetical. Presents the most efficient way to update the progress made after nielsenchuangs textbook was published in 2000.
We outline the general steps of topological quantum computation, as well as discuss various challenges faced by it. Josephson junction arrays, and topological insulators. Sets of topological superconductors containing six elements hexons, all hosting majorana zero modes mzms on the same side of the set, are interfaced with conventional superconductors and semiconductors to perform quantum computation. A topological quantum field theory or topological field theory or tqft is a quantum field theory which focuses on topological invariants although tqfts were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of fourmanifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Topological quantum computation a very basic introduction. Topological quantum computation tqc is the name given to an alternative approach to quantum computation 8 where the elementary computational units are socalled anyonsrather than qubits. Universal topological quantum computation from a superconductorabelian quantum hall heterostructure rsk mong, dj clarke, j alicea, nh lindner, p fendley, c nayak, y oreg.
The hallmark of a topological insulator is the existence of metallic surface states that are higherdimensional analogues of the edge states that characterize a quantum spin hall insulator3,4,5,6. Scalable designs for quasiparticlepoisoningprotected. The nanostructures, with large surfacetovolume ratio and easy fieldeffect gating, provide ideal platforms to detect and manipulate the topological quantum states. We say two knots are topologically equivalent if they can be deformed. Topological quantum computation zhenghan wang microsoft research station q, cnsi bldg rm 2237, university of. A topological dirac insulator in a quantum spin hall phase. We introduce anyons at the systemindependent level of anyon models and discuss the key concepts of protected fusion spaces and statistical quantum evolutions for encoding and processing quantum information. A topological quantum computer is a theoretical quantum computer that employs twodimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a threedimensional spacetime i. The book covers everything required to understand topological faulttolerant. In this sense, this book is complemented by the book introduction to topological quantum computation cambridge university press 2012 written by j. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized nonabelian geometric phases of degenerate ground states. Scalable designs for quasiparticlepoisoningprotected topological quantum computation with majorana zero modes torsten karzig, 1christina knapp,2 roman m.
Get introduction to topological quantum computation 1st edition pdf. Topological quantum computation is a computational paradigm based on. These braids form the logic gates that make up the computer. In this book, the author presents a variety of different topics developed together for the first time, forming an excellent introduction to. Pachos zhenghanwang shors quantum factoring algorithm and the e. Im not sure if these notes will become a book or not. December, 20 one of the main problems with experimental realizations of quantum computation is performing the required operationsresiliently. Inside microsofts quest for a topological quantum computer.
However, unlike ordinary spins, the quantum information associated. Introduction to topological quantum computation by jiannis k. Topological quantum computation topological quantum computation tqc is a computational model built upon systems oftopological phases. This book expands on the authors cbms lectures on knots and. Quantum computation is the eld that investigates the computational power and other properties of computers based on quantum mechanical principles. Topological quantum computation the search for a largescale, errorfree quantum computer is reaching an intellectual junction at which semiconductor physics, knot theory, string theory, anyons, and quantum hall effects are all coming together to produce quantum immunity. The makings of anyonic systems, their properties and their computational power are presented in a pedagogical way. In this paper we examine the problem of algorithmic optimisation in the topological lattice and introduce the required elements that will be needed. Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuit model and to the quantum turing machine model. Therefore, since the physical world is fundamentally quantum mechanical, the foundations of information theory and computer science should be sought in quantum physics. The theory of quantum computation can be constructed from the abstract study of anyonic systems. Quantum computing since democritus by scott aaronson, quantum computation and quantum information by michael a. A suitably designed spin system can support localized excitations quasiparticles that exhibit longrange nonabelian aharonovbohm interactions.
Kitaev, faulttolerant quantum computation by anyons 1997. For readers interested in the physics of topological. The work provides a good reference for quantum computation and quantum information courses, allowing for students to become familiar with major points on the quantum information theoretical aspects of topological quantum computation and the advantages of topological quantum computation for quantum noise resistance. In this book, the author presents a variety of different topics developed together for the. A roadmap for a scalable topological quantum computer. It is called topological because of the particular properties of anyons, that are physically realisable as quasiparticles in topological systems. While this new paradigm holds incredible possibility, quantum computing is very much in its infancy. Cbms regional conference series in mathematics publication year 2010. Schematic of one of the scalable architectures for topological quantum computation proposed by karzig and colleagues. A short introduction to topological quantum computation. Topological quantum computation kitaev 97, flw 00 time.
A gentle introduction eleanor rieffel and wolfgang polak. Pdf introduction to topological quantum computation. Topological quantum computation zhenghan wang microsoft research station q, cnsi bldg rm 2237, university of california, santa barbara, ca 931066105, u. The syllabus, lecture notes and problem sheets will be available as attachments to this page. Mochon, anyons from nonsolvable groups are sufficient for universal quantum computation 2003. Exotic physical properties originating from these topological states endow. Know that ebook versions of most of our titles are still available and may be. Preskill and ogburn, topological quantum computation 1997. Certain exotic states of matter, socalled nonabelian states, have the potential to provide a natural medium for the storage and manipulation of quantum information. Lectures on topological codes and quantum computation a short introductory lecture course on topological codes and quantum computation delivered at the university of innsbruck in may and june 2014.
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