Introduction to topological quantum computation university of leeds. Kitaev, faulttolerant quantum computation by anyons. Measurementonly topological quantum computation work done in collaboration with. Search for nonabelian anyons in topological phases of matter, and build a topological quantum computer. Freedman, larsen, and wang, a modular functor which is universal for quantum computation 2000. 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. In contrast, the \qpresumptive qubitmodel of quantum computation. The computational answer is accessed by bringing anyons together and observing the result. Nontrivial superconductivity in topological mote2 xsx crystals. Wang and kitaev showed in 2002,fklw02, that a topological quantum computer can simulate any computation of a standard quantum computer.
We show that braidings of the metaplectic anyons x. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections. This requires stimulation, manipulation, and observation of topological phenomena at the single quantum particle level, which has not, however, yet been realized. This pedagogical introduction to topological quantum computation includes the following parts. New discovery helps close the gap towards optically.
It is also known type ii 1 factors are determined by their. Topological systems can serve as quantum memories or as quantum computers. This cited by count includes citations to the following articles in scholar. Cas center for excellence in topological quantum computation, university of chinese academy of sciences, beijing, china.
Topological quantum computation aims to achieve this goal by using nonabelian quantum phases of matter. 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. 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. It is one of the competing avenues to demonstrate that quantum computers can complete certain problems that classical computers cannot. Topological quantum computation zhenghan wang microsoft research station q, cnsi bldg rm 2237, university of california, santa barbara, ca 931066105, u. In particular we consider the properties of anyons and their relation to topological quantum computation. Another inspiration for topological quantum computing is faulttolerant quantum computation by anyons 72. Topological quantum computers promise a fault tolerant means to perform quantum computation. 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.
Quantum features may be preserved by being encoded among topological structures of physical evolution systems. Scientists have developed a topological photonic chip to process quantum information, promising a more robust option for scalable quantum computers. In 72, i suggested that topologically ordered states can serve as a physical analogue of errorcorrecting quantum codes. 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.
Discovery of lightinduced switching mechanism advances. Universal quantum computation with metaplectic anyons. American mathematical society with support from the. The braiding and fusion of anyonic excitations in quantum hall electron liquids and 2dmagnets are modeled by modular functors, opening a new. It is believed that the parent materials of many febased superconductors are semimetal in type.
A short introduction to topological quantum computation. American mathematical society, providence, ri 2010 download references. Then topological semimetalbased electronic devices are introduced. 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. Introduction to topological quantum computation by j.
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. A topological quantum computer is a device that relies upon. Pdf ground subspaces of topological phases of matter as. Superconductivity of topological surface states and strong. Quantum computation requires controlled engineering of quantum states to perform tasks that go beyond those possible with classical computers. Introduction to topological quantum computation with non. Jun 30, 2010 topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. Topological quantum computation zhenghan wang ucsb math. On topological quantum computing with mapping class group representations to cite this article. May 16, 2019 quantum computation is any computational model based upon the theoretical ability to manufacture, manipulate and measure quantum states.
The theory of quantum computation can be constructed. These braids form the logic gates that make up the computer. Topological quantum computation with gapped boundaries. Microsoft research station q, cnsi bldg rm 2237, university of. 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. There have been several proposals of quantum computation that are conceptually different, but equivalent to the circuit model. Mathematics of topological quantum computing american.
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. 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. Computers that perform quantum computation are known as quantum computers. Topological quantum computationfrom basic concepts to. Christian schmid, ady stern, david tong, zhenghan wang, harald weinfurter, witlef. Survey paper, to appear in proceedings of contemporary mathematics. 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. Mathematics of topological quantum computing inspire. Larsen, zhenghan wang, topological quantum computation arxiv. 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. They will likely lead to a new wave of technological innovations in communication, computation and cryptography. Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by. Ground subspaces of topological phases of matter as error correcting codes yang qiu and zhenghan wang abstract. May 29, 2010 topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. A measurement of all the qubits at the end of the computation reveals the outcome. Topological quantum computation with gapped boundaries and boundary defects. On topological quantum computing with mapping class group. We conjecture that similar universal anyonic computing models can be constructed for all metaplectic anyon systems sop2 for any odd. One way quantum computation 10 starts from a large entangled.
While this new paradigm holds incredible possibility, quantum computing is very much in its infancy. Topological quantum computation american mathematical society. Such phases allow for quantum information to be stored and manipulated in a nonlocal manner, which protects it from imperfections in the implemented. 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. Collins, \ computing with quantum knots scienti c american, april 2006. Quantum topologyalgebra topological quantum computation models of topological phases of matter work experiences.
First we provide an introduction to anyons and topological models. 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. Topological quantum computing is believed to be inherently faulttolerant. Topological quantum computation is an approach to storing and. Topological quantum computation microsoft research. 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. December, 20 one of the main problems with experimental realizations of quantum computation is performing the required operationsresiliently. Superconductivity in topological semimetals national. 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. Expressing an algorithm in terms of basic quantum gates makes it easy to evaluate its resources and complexity. Academy of science center for excellence in topological quantum computation, university of chinese academy of sciences, 100190 beijing, china edited by allan h.
Download mathematicsof quantumcomputing ebook pdf or read online books in pdf. They underlie the jones polynomial and arise in wittenchern. Quantum computing is the use of quantum mechanical phenomena such as superposition and entanglement to perform computation. 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. 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. The advantage of a quantum computer based on quantum braids over using. The advantage of a quantum computer based on quantum braids. The aim of these lectures is to explain these important results.
Pachos \introduction to topological quantum computation cambridge u. Even considering that the simulation is approximate, given any accuracy, a braid can be found, and it will simulate the computation to that accuracy. Landau institute for theoretical physics, kosygina st. Pdf photonic topological quantum computer george rajna. Pdf introduction to topological quantum computation. Despite a plethora of promising visions towards application and implementation, a number of serious challenges remain. However, enormous scientific and engineering challenges must be overcome for scalable quantum computers to be realized. Submitted on 17 may 2017 v1, last revised 5 dec 2017 this.
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by. Freedman, kitaev, and wang quantph0001071 also showed that a system of anyons can be simulated e. In mathematical terms, these are unitary topological modular functors. Quantum computation is any computational model based upon. Microsoft research, one microsoft way, redmond, wa 98052. That is, any of these models can efficiently simulate any of the others. A topological phase is a class of gapped hamiltonians that realize the same tqft or anyon model in low energy. Larsen, and wang 62 proved that for certain types of nonabelian anyons, braiding enables one to perform universal quantum com. Inside microsofts quest for a topological quantum computer. 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.
I would also like to thank zhenghan wang for answering various ques tions on topics. They underlie the jones polynomial and arise in wittenchernsimons theory. Topological quantum computation topological quantum computation tqc is a computational model built upon systems oftopological phases. Low t c is a limitation for potential application in topological quantum computation. Cas center for excellence in topological quantum computation, university of chinese academy of. Quantum computation and quantum information are of great current interest in computer science, mathematics, physical sciences and engineering.
1060 432 893 537 17 1520 545 257 1083 248 766 273 975 1489 1471 712 545 1061 1206 118 260 200 1118 270 340 304 1239 995 833 1542 328 1545 1644 401 537 433 93 1609 771 111 911 26 1099 1406 1455 89