T OPOLOGICAL MATTER IS FUNDAMENTALLY QUANTUM mechanical, with no correspondent in the classical world. Therefore, it would be quite natural to begin our journey with a review of topics that are particularly relevant to understanding the nature of topological quantum states, such as Berry phases and anyon physics. Yet, even before discussing those topics, pausing to think about some fundamental aspects of quantum theory would be extremely worthwhile. For several decades after the triumph of quantum mechanics in the 1920s and early 1930s, thinking about the foundations of the theory and challenging its standard interpretation were viewed by a majority of physicists as a waste of time, if not plain heresy. The judgment of the physics community on the value of such an endeavor has drastically changed after the groundbreaking work of John Bell and the subsequent experimental tests during the 1970s and 1980s. Today, studying the foundations of quantum mechanics has turned into a flourishing field intimately linked with the explosive rise of quantum information theory. Yet, the subject still has a rather marginal place in standard quantum mechanics textbooks, which focus on the formalism and on the recipes necessary for doing calculations. This chapter is intended for the reader who is less familiar with the problems brought to the fore by quantum information theory and the work on the foundations of quantum mechanics as a brief introduction to these fields and a summary of the basic terminology. But why should one bother about such things when discussing the physics of topological quantum matter? Basically, because the two areas are intimately connected: on the one hand, quantum computation is a prominent item on the list of possible applications for topological quantum states, on the other hand, key concepts in quantum theory, such as quantum entanglement, play a central role in understanding topological matter. Hence, starting the journey with a thought about the foundations of quantum mechanics should enable us to grasp the delicate root that makes the quantum world utterly strange and beautiful, then follow it as it grows out of the sphere of few particle physics into the realm of many-body systems. 1.1 THE QUANTUM MEASUREMENT PROBLEM There is a certain uneasiness about quantum mechanics that every student of this field has probably experienced in some measure. And it is not just the insecurity that results from losing the firm ground of classical intuition; special relativity also deals with this type of problem and does not generate similar feelings. Nor is it at the center of any controversy similar in scale and depth with, for example, the debate [455] between Einstein and Bohr on the foundations of quantum theory. To better understand the source of the problem, let us recall the “standard” way of presenting quantum theory, i.e., starting from certain principles or postulates [126, 440] — a typical approach that can be found in most of the commonly used textbooks. More specifically, let us focus on the key postulates that define the state vectors and their evolution, as well as the observables and the measurements.
T OPOLOGICAL MATTER IS FUNDAMENTALLY QUANTUM mechanical, with no correspondent in the classical world. Therefore, it would be quite natural to begin our journey with a review of topics that are particularly relevant to understanding the nature of topological quantum states, such as Berry phases and anyon physics. Yet, even before discussing those topics, pausing to think about some fundamental aspects of quantum theory would be extremely worthwhile. For several decades after the triumph of quantum mechanics in the 1920s and early 1930s, thinking about the foundations of the theory and challenging its standard interpretation were viewed by a majority of physicists as a waste of time, if not plain heresy. The judgment of the physics community on the value of such an endeavor has drastically changed after the groundbreaking work of John Bell and the subsequent experimental tests during the 1970s and 1980s. Today, studying the foundations of quantum mechanics has turned into a flourishing field intimately linked with the explosive rise of quantum information theory. Yet, the subject still has a rather marginal place in standard quantum mechanics textbooks, which focus on the formalism and on the recipes necessary for doing calculations. This chapter is intended for the reader who is less familiar with the problems brought to the fore by quantum information theory and the work on the foundations of quantum mechanics as a brief introduction to these fields and a summary of the basic terminology. But why should one bother about such things when discussing the physics of topological quantum matter? Basically, because the two areas are intimately connected: on the one hand, quantum computation is a prominent item on the list of possible applications for topological quantum states, on the other hand, key concepts in quantum theory, such as quantum entanglement, play a central role in understanding topological matter. Hence, starting the journey with a thought about the foundations of quantum mechanics should enable us to grasp the delicate root that makes the quantum world utterly strange and beautiful, then follow it as it grows out of the sphere of few particle physics into the realm of many-body systems. 1.1 THE QUANTUM MEASUREMENT PROBLEM There is a certain uneasiness about quantum mechanics that every student of this field has probably experienced in some measure. And it is not just the insecurity that results from losing the firm ground of classical intuition; special relativity also deals with this type of problem and does not generate similar feelings. Nor is it at the center of any controversy similar in scale and depth with, for example, the debate [455] between Einstein and Bohr on the foundations of quantum theory. To better understand the source of the problem, let us recall the “standard” way of presenting quantum theory, i.e., starting from certain principles or postulates [126, 440] — a typical approach that can be found in most of the commonly used textbooks. More specifically, let us focus on the key postulates that define the state vectors and their evolution, as well as the observables and the measurements.
Topology is woven into the subtle fabric of quantum theory. Topological concepts and ideas developed starting in the 1980s by the high energy physics and quantum gravity communities, first viewed by many as nothing but aesthetically appealing abstract possibilities, are central components of modern condensed matter physics and play a key role in the field of quantum information and quantum computation. The discovery of integer and fractional quantum Hall effects and, more recently, of topological insulators and superconductors has brought topological quantum matter in the laboratory, with the perspective of realizing functional platforms for topological quantum computation. Rooted in a solid mathematical formalism and spanning its relevance from fundamental questions in quantum gravity and quantum information to practical aspects regarding new materials and platforms for quantum computation, quantum topology is positioned at a fertile confluence of mathematics, physics, and computer and information science. This book focuses on the aspects of quantum topology that are most relevant from the perspective of condensed matter physics and its relation to quantum computation. We start our journey by sketching the relation between topology and quantum physics and summarizing a few basic mathematical concepts that will be used throughout this work. Beyond dry formal definitions, our main goal is to grasp the delicate root that makes the topological quantum world utterly strange and beautiful and follow it into the realm of condensed matter many-body systems.
T HE FUNDAMENTAL KNOWLEDGE BEHIND MODERN INformation technology, the most visible side of science which permeates almost every aspect of our daily lives, rests on two pillars: information science and quantum mechanics. Knowing how to efficiently communicate and process information and understanding the physical phenomena and material properties that are essential to building functional devices have provided us with unprecedented convenience and mastery over nature, possibly with the price of becoming the slaves of our computers and gadgets. Nevertheless, information science and quantum mechanics enter the backbone of current technologies as distinct and independent fields. Therefore, one is naturally tempted to ask several questions. Is there land between these two realms? If yes, what can grow out of that land? In particular, can information theory tell us something relevant about the foundations of quantum mechanics? On the other hand, can one formulate a quantum information science and in what sense would this be different from the classical version? In this chapter we introduce several basic concepts that will allow us to better understand the implications of these questions and grasp the essential ideas behind the answers formulated so far. Discussing these fundamental concepts will also shed light on the role of topological quantum matter in the economy of quantum computation. Here we focus on the key ideas that were launched into the arena by classical information theory and classical computation science. The metamorphosis of these ideas after the encounter with quantum mechanics will be discussed in the final three chapters of this book.
Abstract At the intersection of physics, mathematics, and computer science, an exciting new field of study has formed, known as “topological quantum.” This research field examines the deep connections between the theory of knots, special types of subatomic particles known as anyons, certain phases of matter, and quantum computation. This book elucidates this nexus, drawing in topics ranging from quantum gravity to topology to experimental condensed matter physics. Requiring only an elementary background in quantum mechanics, this book is appropriate for all readers, from advanced undergraduates to the professional practitioner. The material in presented in a down-to-earth and entertaining way that is far less abstract than most of what is in the literature. While introducing the crucial concepts and placing them in context, the subject is presented without resort to the highly mathematical category theory that underlies the field. This book will be of interest to mathematicians and computer scientists as well as physicists working on a wide range of topics. “Topological quantum” has increasingly been a focus point in the fields of condensed matter physics and quantum information over the last few decades, and the forefront of research now builds on the basic ideas presented in this book. Those interested in working in these field will find this book to be an invaluable introduction as well as a crucial reference.
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
Topological quantum computation has emerged as a rapidly growing research field at the confluence of physics, mathematics, and computer science, bringing together several major themes that fruitfully inform each other. Physics problems, ranging from fundamental questions concerning the foundations of quantum mechanics to problems regarding the classification, realization, and manipulation of topological phases of matter, and mathematical challenges, ranging from identifying the formal tools for the complete classification of topological phases to open problems in computational complexity, converge at this intellectual junction in a combined effort to realize fault-tolerant quantum computation. The convergence of these themes has a significance that goes beyond the simple realization of a certain application – even one as remarkable as a quantum computer – and generates a new paradigm in which the cross fertilization of ideas takes center stage. Thinking about topological quantum matter from the perspective of quantum computation is more creative and fruitful than simply asking the standard questions of many-body condensed matter physics. This is one of the main messages of this book, while the themes involved in this synergy represent its main subject. Therefore, it is fitting to conclude our journey with a chapter on the most explicit manifestation of this new paradigm: topological quantum computation.
We may be on the brink of a quantum transfor -mation driven by our increasing ability to realize, detect, control, and manipulate high-purity quantum states. For decades the quantum world has crept into our lives through modern information technology, the most visible side of science that permeates almost every aspect of our daily conduct. Yet quantumness in its full glory remains a promise for the future, as existing mainstream technologies are essentially based on properties of mixed quantum states realized in, e.g., metal–oxide–semiconductor heterostructures. The scientific backbone of current technologies combines quantum theory with information theory and the theory of computation, everything being sprinkled with a healthy dose of mathematics and brought into reality by remarkable engineering achievements. This multidisciplinary approach has enabled us to understand key physical phenomena and material properties, to communicate and process information efficiently, and, ultimately, to build functional devices that provide us with unprecedented convenience and mastery over nature, possibly with the price of being in danger to become the slaves of our computers and gadgets. However, our technological success is built on a combination of essentially distinct and largely independent scientific fields. Harnessing high-purity quantum states, on the other hand, sets these previously separated “continents of knowledge” on a collision path. The main goal of the last three chapters of this book is to provide a bird's eye view of the “collision zone” and prepare the reader to explore, enjoy, and perhaps exploit the newly raising mountains. We start with a brief introduction to basic concepts in quantum information, which also includes a summary of the operational formulation of quantum mechanics (the “natural” language of quantum information theory) and a synopsis of classical information theory ( Chapter 10 ). For a thorough treatment of concepts, problems, and techniques in quantum information theory the reader is referred to the books by Nielsen and Chuang [ 389 ] and Wilde [ 548 ]. The book by Watrous [ 530 ] focuses on the mathematical theory of quantum information. An undergraduate level introduction to this subject can be found, for example, in Schumacher and Westmoreland [ 458 ]. Some of the philosophical implications of the relationship between quantum mechanics and information theory are discussed in Timpson [ 505 ]. Our survey continues with Chapter 11 , where we discuss a few key aspects pertaining to quantum computation, which can be viewed as the quintessential quantum technology involving the controlled realization and manipulation of high-purity quantum states within a large Hilbert space. Finally, Chapter 12 focuses on topological quantum computation, a promising approach to overcoming the fundamental challenge associated with quantum decoherence based on harnessing the underlying robustness of topological quantum states. This also marks our final return to the realm of topological quantum matter.