Pdf mathematics of quantum computation computational. Mathematics of topological quantum computing inspire. Any reasonable computation can be done by turing machines. Submitted on 17 may 2017 v1, last revised 5 dec 2017 this. Freedman1, alexei kitaev1, zhenghan wang 2 1 microsoft research, one microsoft way, redmond, wa 980526399, usa 2 indiana university, dept. A topological quantum computer is a theoretical quantum computer that employs. Topological quantum computation microsoft research. The talk and these notes are almost entirely based on the lecture notes of john preskill. Condensed matter physics and topological field theory. In mathematical terms, these are unitary topological modular functors. Iris cong meng cheng and zhenghan wang topological quantum.
We study faulttolerant quantum computation with gapped boundaries. Topological quantum computation zhenghan wang ucsb math. Topological quantum computation american mathematical society. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by. Topological quantum computation zhenghan wang microsoft research station q, cnsi bldg rm 2237, university of california, santa barbara, ca 931066105, u. Topological quantum computation tqc has recently attracted lots of. 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. Topological phases of matter a topological quantum phase is represented by a quantum theory whose low energy physics in the thermodynamic limit is modeled by a. Simulation of topological field theories by quantum computers michael h. In topological quantum computing, information is encoded in knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay.
Pachos zhenghan wang 1 quantum information processing volume 11, pages 1969 1973 2012 cite this article. Regional conference series in mathematics i conference board of the mathematical sciences volume 112 zhenghan wang topological quantum computation 2010, american mathematical soc. Iris cong and zhenghan wang, topological quantum computation with. 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 nonabelian anyons, meaning that they obey nonabelian braiding. Microsoft has been working on topological quantum computing for more than a decade, for most of which the necessary qubits were hypothetical. In topological quantum computation one aims to make use of quantum systems described by topological quantum field theory for quantum computation for the time being see at quantum computation for more related concepts. Landau institute for theoretical physics, kosygina st.
Larsen, zhenghan wang, topological quantum computation arxiv. Topological quantum computation and quantum logic zhenghan wang microsoft station q uc santa barbara. Introduction to topological quantum computation by j. Research interests quantum topologyalgebra, quantum computation publications and slides. A modular functor which is universal for quantum computation michael h. Topological quantum computation freedman 97, kitaev 97, flw 00.
Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. Lukasz cincio and guifre vidal, characterizing topological order by studying the ground states on an infinite cylinder phys. Quantum topologyalgebra topological quantum computation models of topological phases of matter work experiences. Iris cong meng cheng and zhenghan wang topological. 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 nonabelian anyons, meaning that they obey nonabelian braiding statistics. The braiding and fusion of anyonic excitations in quantum hall electron liquids and 2dmagnets are modeled by modular functors, opening a new. Simulation of topological field theories by quantum computers. They underlie the jones polynomial and arise in wittenchernsimons theory.
A topological phase is a class of gapped hamiltonians that realize. Microsoft research, one microsoft way, redmond, wa 98052. Modeling and classification of topological phases of matter. However, enormous scientific and engineering challenges must be overcome for scalable quantum computers to be realized. Kitaev, faulttolerant quantum computation by anyons. The boundaries and twist defects of the color code. Kesselring 1, fernando pastawski 1, jens eisert 1, and benjamin j. Microsoft research station q, cnsi bldg rm 2237, university of.
They will likely lead to a new wave of technological innovations in communication, computation and cryptography. Topological quantum computation with gapped boundaries. In 72, i suggested that topologically ordered states can serve as a physical analogue of errorcorrecting quantum codes. By exploring the remarkable quasiparticle excitations of the color code, we describe and classify all of its boundaries and topological defects. Simulation of topological field theories by quantum. Search for nonabelian anyons in topological phases of matter, and build a topological quantum computer. Leakage into a noncomputational subspace seems to be unavoidable for universality. The braiding and fusion of anyonic excitations in quantum hall electron liquids and 2dmagnets are modeled by modular functors. A modular functor which is universal for quantum computation. In topological quantum computation one aims to make use of quantum systems described by topological quantum field theory for quantum computation. Download pdf mathematicsofquantumcomputing free online.
Topological quantum computation topological quantum computationtqc is a computational model built upon systems oftopological phases. Seminars and conferences weekly geometry, topology and physics seminar. The computational answer is accessed by bringing anyons together and observing the. Alexei kitaev, michael larsen, zhenghan wang, topological quantum computation, bulletin of the.
American mathematical society with support from the. These braids form the logic gates that make up the computer. Cbms regional conference series in mathematics publication year 2010. The subject of quantum computing brings ideas from classical information theory, computer. They underlie the jones polynomial and arise in wittenchern. Theoretically, we would like to classify topological phases of matter, and experimentally, find nonabelian objects in nature. As described above, topological quantum computation is novel wa y to perform quantum computation which attempts to solv e the problem of decoherence at the chapter 1. Quantum computation and quantum information are of great current interest in computer science, mathematics, physical sciences and engineering.
A short introduction to topological quantum computation. Another inspiration for topological quantum computing is faulttolerant quantum computation by anyons 72. Additional material for the book american mathematical society. Finiteenergy topological quasiparticle excitationsanyons quasiparticles a, b, c a b c. Sep 21, 2012 introduction to topological quantum computation by j. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized nonabelian geometric phases of degenerate ground states.
On topological quantum computing with mapping class group representations wade bloomquist and zhenghan wang abstract. Topological quantum computation is based on the possibility of the realization of some tqfts in nature as topological phases of quantum matter. Promising models for faulttolerant quantum computation are based on topological phases of matter, and of particular interest is the color code model, due to its numerous symmetries. Quantum computers should be capable of performing tasks that would be very difficult, if not impossible, with digital computers, such as finding the prime factors of large numbers, searching large databases, and simulating quantum systems. We rst present the algebraiccategorical structure of gapped boundaries and boundary defects.
In physics, topological order is a kind of order in the zerotemperature phase of matter also known as quantum matter. Scott aaronson, quantum computing since democritus, cambridge. For the time being see at quantum computation for more. Microscopically, topological orders correspond to patterns of longrange quantum. Topological phases local hilbert space local, gapped hamiltonian e e gap two gapped hamiltonians 1, 2 realize the same topological phase of matter if there exists a continuous path connecting them without closing the gapa phase transition. The computational answer is accessed by bringing anyons together and observing the result. Larsen, and zhenghan wang proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do, and vice versa. Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. Larsen, and zhenghan wang proved that a topological quantum computer. We propose an encoding for topological quantum computation utilizing quantum representations of mapping class groups. The boundaries and twist defects of the color code and their applications to topological quantum computation.
The advantage of a quantum computer based on quantum braids over using. Pdf nonabelian anyons and topological quantum computation. Topological quantum computation zhenghan wang publication year. Freedman1, michael larsen2, zhenghan wang 2 1 microsoft research, one microsoft way, redmond, wa 980526399, usa 2 indiana university, dept. Mathematics of topological quantum computing american. Download pdf 588 kb abstract we survey some recent work on topological quantum computation with gapped boundaries and boundary defects and list some open problems. The theory of quantum computation can be constructed from the abstract study of anyonic systems.
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