Introduction to topological quantum matter and quantum computation, by tudor d. Sep 12, 2019 for the love of physics walter lewin may 16, 2011 duration. There is an increasing interest in realizing topological quantum phases in synthetic quantum systems 15, including ultracold atoms in optical lattices, rydberg atoms, trapped ions, and superconducting qubits. Provides a comprehensive introduction to topological quantum codes and faulttolerant quantum computation with them. The topology of matter the world of quantum matter. Tpm umtc tqc therefore the practical aspect of topological quantum computation hinges on the existence of nonabelian topological states. Aug 26, 2019 one of the recurring central themes in physics is the search for exotic phases of matter stemming from strong correlations between constituent particles. Topological quantum matter topologically protected matter unique groundstate protected by a topological invariant chern number, z 2index.
Were quantum information encoded in topological properties of matter, and were the quantum gates dependent only on the topology of the evolutions, then both should be inherently protected from local perturbations. Nonabelian anyons and topological quantum computation s. 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 systems can serve as quantum memories or as quantum computers. Operators o that belong to the normalizer of s leave invariant the code space c. Topological quantum computation microsoft research. Manybody topological invariants from randomized measurements. Presents the most efficient way to update the progress made after nielsenchuangs textbook was published in 2000. Topological quantum computing lukas hahn 1 introduction1 2 qubits and universality1. We introduce anyons at the systemindependent level of anyon models and. Such topological quantum computation would exhibit in.
The discovery of topological insulators has established a standard paradigm about the topological quantum states of matter, which includes band inversion, bulkboundary correspondence, the relationship between symmetry and topological invariance, etc. Topological quantum computers promise a fault tolerant means to perform quantum computation. Introduction to topological quantum computing steven h. Quantum computing exploits quantum mechanical phenomena such as superposition and entanglement to perform operations on data, which in many cases, are infeasible to do efficiently on. Introduction to topological quantum computation by jiannis. Pdf introduction to topological quantum computation. Topological phases of matter and quantum computation. 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. In topological quantum computing, information is encoded in knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. These braids form the logic gates that make up the computer. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized nonabelian geometric phases of degenerate ground states.
Topological quantum computing tqc is a theoretical model for quantum computation. Topological orders and quantum orders extend and deepen our previous understanding of orders in states of matter, and guide us to discover new states of matter. This book presents a theoretical description of topological matter in terms of effective field theories, and in particular topological field theories, and gives an introduction to both topological order and symmetryprotected phases using a modern mathematical language. This is a book that will show you even new to old thing. Topological precision has been confirmed in quantum hall liquids by experiments to an accuracy of 1010, and harnessed to stabilize quantum memory. Jul 20, 2017 it is, therefore, a revolutionary theory of topological quantum chemistry, a description of the universal global properties of all possible band structures and materials. Pachos \introduction to topological quantum computation cambridge u. A new type of quantum computer, known as a topological quantum computer, has been proposed by theoretical physicists. The world of quantum matter aims to introduce this branch of science, covering the fundamentals through to the most cuttingedge topics. First we provide an introduction to anyons and topological models.
In physics, topological order is a kind of order in the zerotemperature phase of matter also known as quantum matter. The makings of topological systems, their properties and their computational power are presented in a pedagogical way. 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. What is quantum computation and what makes it different from classical computation. Connections between topological and gapless ultraquantum matter, such as that discussed in the ultraquantum metals session, were also drawn.
Introduction to topological quantum computation with nonabelian anyons arxiv. The synthesis of the two ideas ushered in topological quantum computation. Topological quantum matter a field theoretical perspective. Quantum computation with topological codes springerlink. In this approach, information is stored in the lowest energy states of manyanyon systems and processed by braiding nonabelian anyons. The advantage of a quantum computer based on quantum braids over using. Topological quantum computing kyle evitts introduction quantum computers have generated a lot of interest in the academic world, security circles, and tech companies because of their computational advantages over classical computers. Classification of topological quantum matter with symmetries. The existence of topological phases of matter with nonabelian anyons would lead us to topological quantum computation via unitary modular tensor categories. He is known particularly for his theoretical discovery of perfect conducting 1d chiral luttinger liquids, z2 spin liquid with emergent. The study of quantum matter falls into the field of condensed matter physicsa fundamental branch of physicswhich is the study of the structure and behaviour of solid and fluid matter. However, even though we have a good understanding of the mathematical theory. Introduction to topological quantum matter and quantum. This pedagogical introduction to topological quantum computation includes the following parts.
Majorana fermions are a fascinating medium for discovering new phases of matter. To understand this, we will discuss exactly what a world line is, how. 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. In this book, the author presents a variety of different topics developed together for the first time, forming an excellent introduction to. Amazingly, those quasiparticles in two spacial dimensions do not necessarily obey the standard bose or fermi statistics. Despite a plethora of promising visions towards application and implementation, a number of serious challenges remain. Topological quantum computation is a computational paradigm based on topological phases of matter, which are governed by topological quantum field theories. In the last twenty years, several themes have come to the forefront of quantum condensed matter physics research through crossfertilization with other disciplines, such as. Pdf a short introduction to topological quantum computation. Topological quantum phases, originally discovered and investigated in condensed matter physics, have. Anomalies play an important role in topological phases, with surface properties often connected to different kinds of anomaly cancellation between the surface and the bulk. In particular we consider the properties of anyons and their relation to topological quantum computation. Even though there is vast literature on these subjects, the book fills an important gap by providing a concise introduction to both topological order and symmetryprotected phases using a modern mathematical language, and developing the theoretical concepts by highlighting the physics and the physical properties of the systems.
Introduction to topological quantum computation with anyons chiahsun paul lee abstract. Topological quantum computing heidelberg university. 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. 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 short introduction to topological quantum computation. Mathematics of topological quantum computing microsoft.
As already spoiled in the introduction, we want to construct a computer device out. The evolution of a theory in 1928, just two years after the formulation of quantum mechanics, the german physicist arnold sommerfeld modified the classical freeelectron 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. Topological states of matter are a particular class of nonlandau states, which are characterized by the notion of topological order. The computational answer is accessed by bringing anyons together and observing the result. Combining physics,mathematicsandcomputerscience,topological quantumcomputation is a rapidly expanding research area focused on the exploration of quantum evolutions that are immune to errors. Quantum information meets quantum matter springerlink. Quantum computation with topological codes from qubit to topological faulttolerance. Both the encoding and the processing are inherently resilient against. 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. Quantum computation david divincenzo it is the prospect of building a quantum computer, rather than the fascinating properties of quantum physics or of entanglement, that is responsible for much of todays interest in quantum information. Introduction to topological quantum computation with non. In this book, the author presents a variety of different topics developed together for the. One of the recurring central themes in physics is the search for exotic phases of matter stemming from strong correlations between constituent particles.
A longwinded introduction on the history of this field the original string theorist 1867. Collins, \computing with quantum knots scienti c american, april 2006. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This volume contains the proceedings of the ams special session on topological phases of matter and quantum computation, held from september 2425, 2016, at bowdoin college, brunswick, maine.
Quantum computation with topological codes from qubit to. Andrei bernevig 3,6 8 9 for the past century, chemists and physicists have advocated funda mentally different perspectives on materials. The advantage of a quantum computer based on quantum braids. Mele, and shoucheng zhang has, however, uncovered a new electronic phase of matter called a topological insulator, in which unusual topological properties are protected by symmetry. For example in the fractional quantum hall effect, the topological order is directly responsible for the celebrated properties of fractional charge, anionic statistics and gapless chiral edge modes.