Topological quantum computation american mathematical society. We show that braidings of the metaplectic anyons x. We pursue a totally different approach based on topological quantum field theories tqfts, and prove that a lattice implementation of the disk axiom and annulus axiom in tqfts as essentially the equivalence of tqo1 and tqo2 conditions in bhm10. Another inspiration for topological quantum computing is faulttolerant quantum computation by anyons 72. Topological quantum computation with gapped boundaries and boundary defects.
Among these, topological states have become a major research direction in the past decade, from quantum spin liquids, to topological insulators and superconductors, to examples in photonics and mechanical systems. They underlie the jones polynomial and arise in wittenchern. Mathematics of topological quantum computing american. 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. Wang and kitaev showed in 2002,fklw02, that a topological quantum computer can simulate any computation of a standard quantum computer.
Despite a plethora of promising visions towards application and implementation, a number of serious challenges remain. Introduction to topological quantum computation university of leeds. Scientists have developed a topological photonic chip to process quantum information, promising a more robust option for scalable quantum computers. In contrast, the \qpresumptive qubitmodel of quantum computation. Topological quantum computation is an approach to storing and. First we provide an introduction to anyons and topological models. Superconductivity in topological semimetals national. Pdf introduction to topological quantum computation. Search for nonabelian anyons in topological phases of matter, and build a topological quantum computer. These braids form the logic gates that make up the computer. Microsoft research, one microsoft way, redmond, wa 98052. 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. Topological quantum computation aims to achieve this goal by using nonabelian quantum phases of matter.
I would also like to thank zhenghan wang for answering various ques tions on topics. Even considering that the simulation is approximate, given any accuracy, a braid can be found, and it will simulate the computation to that accuracy. This requires stimulation, manipulation, and observation of topological phenomena at the single quantum particle level, which has not, however, yet been realized. Introduction to topological quantum computation with non. 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. As quantum technologies advance, we get closer to finding solutions to some of the worlds most challenging problems. Landau institute for theoretical physics, kosygina st. Low t c is a limitation for potential application in topological quantum computation. It is one of the competing avenues to demonstrate that quantum computers can complete certain problems that classical computers cannot.
The advantage of a quantum computer based on quantum braids over using. It is also known type ii 1 factors are determined by their. American mathematical society, providence, ri 2010 download references. Jun 30, 2010 topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories.
The use of topology to protect quantum information is well known to the condensedmatter community and, indeed, topological quantum computing is a burgeoning field of research. The braiding and fusion of anyonic excitations in quantum hall electron liquids and 2dmagnets are modeled by modular functors, opening a new. This cited by count includes citations to the following articles in scholar. Pachos \introduction to topological quantum computation cambridge u.
Finally, we discuss several important aspects that should receive great effort in the future, including controllable synthesis, manipulation of quantum states, topological field effect transistors, spintronic applications, and topological quantum computation. Topological quantum computers use particles with exotic exchange statistics called nonabelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the fibonacci anyon model. Mathematics of topological quantum computing article pdf available in bulletin of the american mathematical society 552 may 2017 with 407 reads how we measure reads. New discovery helps close the gap towards optically. 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. 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. The aim of these lectures is to explain these important results. Topological quantum computation with gapped boundaries. Superconductivity of topological surface states and strong.
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. Quantum features may be preserved by being encoded among topological structures of physical evolution systems. Microsoft research station q, cnsi bldg rm 2237, university of. Topological quantum computation zhenghan wang microsoft research station q, cnsi bldg rm 2237, university of california, santa barbara, ca 931066105, u. However, enormous scientific and engineering challenges must be overcome for scalable quantum computers to be realized. 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. Freedman, kitaev, and wang quantph0001071 also showed that a system of anyons can be simulated e. Cas center for excellence in topological quantum computation, university of chinese academy of. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by. Survey paper, to appear in proceedings of contemporary mathematics. This pedagogical introduction to topological quantum computation includes the following parts. Quantum topologyalgebra topological quantum computation models of topological phases of matter work experiences. Topological computation initialize create particles apply operators braid output measure.
A measurement of all the qubits at the end of the computation reveals the outcome. Cas center for excellence in topological quantum computation, university of chinese academy of sciences, beijing, china. Topological quantum computationfrom basic concepts to. It is believed that the parent materials of many febased superconductors are semimetal in type. May 29, 2010 topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections. Submitted on 17 may 2017 v1, last revised 5 dec 2017 this. Computers that perform quantum computation are known as quantum computers. They underlie the jones polynomial and arise in wittenchernsimons theory. In mathematical terms, these are unitary topological modular functors. Discovery of lightinduced switching mechanism advances.
The theory of quantum computation can be constructed from the abstract study of anyonic systems. 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. Then topological semimetalbased electronic devices are introduced. Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. The computational answer is accessed by bringing anyons together and observing the result.
There have been several proposals of quantum computation that are conceptually different, but equivalent to the circuit model. Topological quantum computation using majorana fermions. Pdf ground subspaces of topological phases of matter as. 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. Measurementonly topological quantum computation work done in collaboration with.
In particular we consider the properties of anyons and their relation to topological quantum computation. Quantum computation and quantum information are of great current interest in computer science, mathematics, physical sciences and engineering. The mechanism establishes a new way to control the topological material, driven by backandforth motion of atoms and electrons, which will enable topological transistor and quantum computation. Download mathematicsof quantumcomputing ebook pdf or read online books in pdf. Topological quantum computation microsoft research.
That is, any of these models can efficiently simulate any of the others. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by braiding nonabelian anyons. Expressing an algorithm in terms of basic quantum gates makes it easy to evaluate its resources and complexity. However, scientists who study these topological materials face a challengehow to establish and maintain control of these unique quantum behaviors in a way that makes applications like quantum computing possible. Topological quantum computers promise a fault tolerant means to perform quantum computation. A topological quantum computer is a device that relies upon. Topological quantum computation topological quantum computation tqc is a computational model built upon systems oftopological phases. Mathematics of topological quantum computing inspire. 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. Freedman, larsen, and wang, a modular functor which is universal for quantum computation 2000.
Pdf photonic topological quantum computer george rajna. Collins, \computing with quantum knots scienti c american, april 2006. Christian schmid, ady stern, david tong, zhenghan wang, harald weinfurter, witlef. Ground subspaces of topological phases of matter as error correcting codes yang qiu and zhenghan wang abstract. 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. In 72, i suggested that topologically ordered states can serve as a physical analogue of errorcorrecting quantum codes. A topological phase is a class of gapped hamiltonians that realize the same tqft or anyon model in low energy. In this experiment, wang and his colleagues demonstrated that control by using light to steer quantum states in a dirac semimetal.
They will likely lead to a new wave of technological innovations in communication, computation and cryptography. Nontrivial superconductivity in topological mote2 xsx crystals. On topological quantum computing with mapping class group representations to cite this article. While this new paradigm holds incredible possibility, quantum computing is very much in its infancy. May 16, 2019 quantum computation is any computational model based upon the theoretical ability to manufacture, manipulate and measure quantum states. Universal quantum computation with metaplectic anyons. Topological quantum computationfrom basic concepts to first. On topological quantum computing with mapping class group. Key laboratory of artificial structures and quantum control ministry of education, shenyang national laboratory for materials science, school of physics and astronomy, shanghai jiao tong university, shanghai, 200240 china. The advantage of a quantum computer based on quantum braids.
A short introduction to topological quantum computation. Topological quantum computation zhenghan wang ucsb math. Collins, \ computing with quantum knots scienti c american, april 2006. The theory of quantum computation can be constructed. Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories.
Academy of science center for excellence in topological quantum computation, university of chinese academy of sciences, 100190 beijing, china edited by allan h. We conjecture that similar universal anyonic computing models can be constructed for all metaplectic anyon systems sop2 for any odd. Quantum computation is any computational model based upon. Topological systems can serve as quantum memories or as quantum computers. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented. American mathematical society with support from the. Freedman, kitaev, and wang, simulation of topological field theories by quantum computers 2000. Topological quantum computing is believed to be inherently faulttolerant.
Larsen, zhenghan wang, topological quantum computation arxiv. Introduction to topological quantum computation by j. One way quantum computation 10 starts from a large entangled. Kitaev, faulttolerant quantum computation by anyons.
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