Non abelian topological systems a fascinating property of many topological phases are collective fractionalized excitations which are highly non local in nature. Here are some of the handwritten notes that have not yet been typeset. Iii, we derive the geometric phase in open systems, in both the abelian berry phase and nonabelian cases. Quantum science and technology paper introduction to topological quantum computation with non abelian anyons to cite this article. A twodimensional medium that supports abelian anyons has a topological. We apply this methodology for configuration spaces of graphs. Topological quantum computers use particles with exotic exchange statistics called non abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the fibonacci anyon model. Abelian anyons non abelian anyons physical realization of non abelian anyons fractional quantum hall effect geometric phase topological quantum computer 3. Abelian hidden subgroup problem, discrete logarithm 2009 handwritten notes see also notes on nonabelian hsp quantum searching 2009 handwritten notes see also 2009 notes on quantum lower bounds. In theoretical physics, quantum chromodynamics qcd is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. What exactly are anyons and how are they relevant to.
Topological quantum computation based on chiral majorana. Each chapter was covered in a lecture of 2 45 minutes, with an additional 45minute lecture for exercises and homework. Topological quantum computationfrom basic concepts to. Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. A universal quantum computer can be constructed using abelian anyons. Topological quantum computation from nonabelian anyons galileo. Quantum shannon theory, 90 pages updated january 2018. From nonabelian anyons to quantum computation to coin. Topological quantum computation purdue engineering. These are realized in terms of the nonabelian berry connection and are obtained by driving the control parameters. Quantum computing with nonabelian quasiparticles request pdf. Pdf nonabelian berry connections for quantum computation. Introductory quantum mechanics good coverage, explanations medium. Introduction to topological quantum computation with nonabelian anyons.
Abelian and nonabelian geometric phases in adiabatic open. Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Dummys guide to topological quantum computing uses 2 dimensional systems with nonabelian statistics. Introduction to topological quantum computation with non abelian anyons bernard field and tapio simula school of physics and astronomy, monash university, victoria 3800, australia dated. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as \\it non abelian anyons, meaning that they obey \\it non abelian braiding statistics.
Topological quantum computation from nonabelian anyons. Quantum information processing is the result of using the physical reality that quantum theory tells us about for the purposes of performing tasks that were previously thought impossible or infeasible. Topological quantum computers promise a fault tolerant means to perform quantum computation. Lectures on quantum mechanics graduate level textbook. Topological quantum computation schemes where quantum information is stored non locally provide, in theory, an elegant way of avoiding the deleterious effects of decoherence, but they have. In the second half of this thesis, a scheme for processing quantum information is presented, which uses non abelian. Microsoft research is building a quantum computer powered. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner. Developing a topological quantum bit has been a longterm dream for man and non abelian systems could provide the answer. The associated unitary monodromy braiding matrices become the required logic gates in the special case when the vortex spin is s 14, which corresponds to the case of ising nonabelian anyons, found in di. Dec 15, 2014 topological quantum computing tqc is a newer type of quantum computing that uses braids of particle tracks, rather than actual particles such as ions and electrons, as the qubits. Anyonic quantum computation might be realized using quasiparticles of the fractional quantum hall. Qcd is a type of quantum field theory called a non abelian gauge theory, with symmetry group su3.
Fibonacci quasiparticle could form basis of future quantum. Quantum mechanics very clear explanations, doesnt cover everything. Abelian and non abelian geometric phases in adiabatic open quantum systems m. Not all anyons are directly useful in topological quantum computation tqc. However, the online articles including wikipedia seem unusually vague and impenetrable as far as explaining topological quantum computing and anyons goes. Topological quantum computers use particles with exotic exchange statistics called nonabelian anyons, and the simplest anyon model which allows for. Topological quantum materials for realizing majorana.
The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but a re particles known as non abelian anyons, meaning that they obey non abelian braiding statistics. Topological quantum computation the properties of non abelian states that are importantforourdiscussionare thequantumdimensions of the anyons, the unitary transformations that they generate by braiding, and their fusion rules. Again, bringing forward the familiar, we note that to say the braiding transformations of the wave functions are non abelian is simply to say they are non commutative. Pdf developing a robust approach to implementing non. Kitaevs topological quantum computing scheme caused great excitement because a particle that was strongly suspected to be a non abelian anyon already existed.
I will not be as precise as mathematicians usually want. Quantum computing qubits quantum decoherence noise extremely fragile need to be shielded from environmental perturbations topological quantum computing topological qubits based on non abelian anyons these twodimensional quasiparticles form braids which are much more stable. Topological quantum computation is an approach to storing and manipulating quantum infor. We find that for topological quantum field theories tqfts, where the combinatorial nature of the propagation allows a complete analysis, hardness and noncommutativity are tightly linked. Introduction to topological quantum computation with nonabelian. Topological quantum computers use particles with exotic exchange statistics called nonabelian anyons, and the simplest anyon model which. Related content braiding operators are universal quantum gates louis h kauffman and samuel j. Non abelian anyons are special as they are thought to be able to process. Quantum information processing 2017 hosted by microsoft. Devices that perform quantum information processing are known as quantum computers. These notes cover several major topics in quantum algorithms, divided into six parts. Quantum computing qubits quantum decoherence noise extremely fragile need to be shielded from environmental perturbations topological quantum computing topological qubits based on non abelian anyons these twodimensional quasiparticles form braids which are.
It was an elusive entity that had been discovered a decade earlier by a graduate student at the massachusetts institute of technology doing his first set of experiments bob willett. Qip 2017 was hosted in seattle, washington by the quarc group at microsoft research in redmond and the university of washington the international annual quantum information processing qip series is the premier meeting for theoretical quantum information research. There are many proposals for building a quantum computer, for example. On the another hand, when it comes to the realization of quantum computers in the applications of topological theory for anyons, the. Nonabelian berry connections for quantum computation. Certain species of anyons called nonabelian give rise to an exchange. Here we show that there is much more to quantum computing by exploring some of the many quantum algorithms that have been developed over the past twenty years. Microsoft, the only big company investing in topological quantum computing, has been working to create non abelian anyons in materials since 2005, when the company established station q, a.
I have been trying to get a basic idea of what anyons are for the past couple of days. Erez berg topological superconductors majorana zero modes and their physical realizations duration. Topological quantum computers use particles with exotic exchange statistics called non abelian anyons. Non abelian anyons are a controversial part of quantum computing, though. Majorana anyons, nonabelian statistics and quantum. Dummys guide to topological quantum computing uses 2 dimensional systems with non abelian statistics. Graph design for secure multiparty computation over non.
Majorana zero modes and topological quantum computation npj. Pdf introduction to topological quantum computation with non. These lecture notes were formed in small chunks during my \ quantum computing course at the university of amsterdam, febmay 2011, and compiled into one text thereafter. 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. Each chapter was covered in a lecture of 2 45 minutes, with an additional 45minute lecture for. They are exploited as hardware platforms for faulttolerant topological quantum computing. Pdf nonabelian anyons and topological quantum computation. Two physicists bet over a quantum computing moon shot wired.
The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as \\it non abelian anyons, meaning that they obey \\it non abelian. Abstract for topological quantum computing for beginners, by john preskill i will describe the principles of faulttolerant quantum computing, and explain why topological approaches to fault tolerance seem especially promising. Pdf topological quantum computationfrom basic concepts to. Quantum computation with abelian anyons springerlink. Majorana zero modes and topological quantum computation. These carry fractions of the quantum numbers of the underlying microscopic degrees of freedom. These braids form the logic gates that make up the computer. Nonabelian anyons and topological quantum computation. In order for such quantum protocols to be viable, though, a number of practical obstacles involving the communication and processing of quantum information must be resolved. The future of quantum computing could depend on this. Developing a robust approach to implementing non abelian anyons and topological quantum computing in a modified kitaev honeycomb lattice model. The advantage of a quantum computer based on quantum. We believe that addressing the hsp in the non abelian case is the most important challenge at present in quantum computing.
Introduction to topological quantum computation with non. A gentle introduction eleanor rieffel and wolfgang polak. The existence of topological phases of matter with non abelian anyons would lead us to topological quantum computation via unitary modular tensor categories. Fibonacci quasiparticle could form basis of future. This relation is rooted in the fact that the basic playground of qprocessing is the complex space hn generated by the set bnof binary vectors of length n, which is the basic arena of classical computation. Majorana modes, nonabelian anyons, and topological quantum. Oct 23, 2018 the main goal of our proposal is to show that the chiral majorana fermion edge state of the tsc can be used to realize non abelian quantum gate operations on electron states, even if there is no non abelian anyon traveling along the edge. Quantum information is encoded in nonlocal topological degrees of freedom that do not couple to any local quantity. Im typing as we go so please forgive all typos and unclear parts etc. Nonabelian statistics and topological quantum information. To implement faulttolerant topological quantum computing, we seek very speci. The concept was put forward by alexei kitaev in 1997, and was met with skepticism. More broadly, we propose that a physical system s with a non abelian topological term in its lagrangian may have observables that are nphard or.
April 23, 2018 topological quantum computers promise a fault tolerant means to perform quantum computation. A short introduction to topological quantum computation. Topological quantum computing tqc is a newer type of quantum computing that uses braids of particle tracks, rather than actual particles such as ions and electrons, as the qubits. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as it non abelian anyons, meaning that they obey it non abelian braiding statistics. Majorana modes, nonabelian anyons, and topological. Nonabelian quantum statistics on graphs springerlink. We describe recent work showing how to find braids which can be used to perform arbitrary quantum computations using a specific kind of quasiparticle those described by the socalled fibonacci anyon model which are thought to exist in the experimentally observed. May 16, 2019 what is topological quantum computing, where is the field heading into, what is the current state of the art researchin this blog, which was part of uc berkeleys quantum computing course. Despite the theoretical progress made during the past decade on using mzms in universal quantum computation 1417, due to the localized and pointlike nature of mzms, all existing proposed archi. States are usually manipulated by dragging braiding quasiparticles around each other. Pdf introduction to topological quantum computation. Aug 11, 2016 institute for quantum computing 29,660 views 2. Topological quantum computation schemes where quantum information is stored nonlocally provide, in theory, an elegant way of avoiding the deleterious effects of decoherence, but they have.
Topological quantum computation aims to achieve this goal by using nonabelian quantum phases of matter. Qft, qat, algebraic quantum field theory, quantum field theories, nonabelian algebriac topology applications. Nonabelian anyons were proposed as elementary excitations of the. Experimental and theoretical successes have made us take a close look at quantum. Introduction to topological quantum computation theoretical. Tpm umtc tqc therefore the practical aspect of topological quantum computation hinges on the existence of non abelian topological states. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non abelian anyons on networks that are used in quantum computing. Quantum computing qubits quantum decoherence noise extremely fragile need to be shielded from environmental perturbations topological quantum computing topological qubits based on nonabelian anyons these twodimensional quasiparticles form braids which are much more stable. Different non abelian systems differ in these properties. Beside its intrinsic mathematical interest, the importance of this problem is enhanced by the fact that it contains as special case the graph isomorphism problem.
Nonabelian anyonsparticles whose exchange noncommutatively transforms a systems quantum stateare widely sought for the exotic fundamental physics they harbour and for quantum computing. In this thesis, we primarily study the non topological decoherence ef fects existing in realistic. Non abelian superconductors are novel systems with exotic quasiparticle excitations, namely majorana fermions, which obey non abelian quantum statistics. Singleanyon operations such as hopping from site to site on a lattice suffice to perform all quantum logic operations. Anyonic quantum computation might be realized using quasiparticles of the fractional quantum. Topological quantum computation aims to achieve this goal by using non abelian quantum phases of matter.
A gentle introduction eleanor rieffel and wolfgang polak the mit press cambridge, massachusetts london, england. Two qubit quantum logic gates such as controllednot operations are performed using topological effects. Graph design for secure multiparty computation over non abelian groups xiaoming sun 1, andrew chichih yao, and christophe tartary,2 1 institute for theoretical computer science tsinghua university beijing, 84 peoples republic of china. Hansson anyon school berlin, 20 fractional quantum statistics t.
425 272 198 435 1229 265 1606 1118 796 1010 1315 1559 1494 1325 8 120 1122 260 37 64 1022 111 1007 1448 1629 1273 1109 1405 406 1052 149 1595 762 1388 210 618 592 451 840 285 795 501 317