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arXiv:1705.06206 · Bulletin of the American Mathematical Society

Mathematics of topological quantum computing

Eric C. Rowell, Zhenghan Wang

Published Jan 16, 2018• 87 citations
arXivHTMLPDF

Abstract

In topological quantum computing, information is encoded in “knotted” quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of 10 − 10 10^{-10} and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics, and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.

DOI
10.1090/bull/1605

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In topological quantum computing, information is encoded in “knotted” quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of 10 − 10 10^{-10} and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics, and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.
In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.
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.
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.
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.
Topological phases of matter have been the focus of intense research for the past decades. Their intrinsic topological nature exhibits perturbation-resilient effects in macroscopic materials. These properties have applications that span sensing, electronics, spintronics, lasers, and quantum computing. They have also inspired the development of new mathematical models to study their underlying topological quantum field theory. In this work, we visit three aspects of topological phases of matter. First, we use tensor-network tools, along with the aid of Purdue's high-performance computing clusters, to simulate symmetry-protected topological phases in 1D. We update works on the realization of the Haldane phase in Cu2_22​O excitons. In addition, we related the group-cohomological classification of these phases to their order parameters. Second, we design 2D topological insulators with higher Chern numbers. This is achieved through distant physical interactions in accordance with algebraic geometry arguments. Third, we study 2D topological quantum error-correcting codes. In particular, we construct the boundaries, domain walls, and 000D defects of Abelian twisted quantum doubles leveraging anyon condensation. We discuss these codes' advantages in correcting errors for quantum computing. We also give microscopic and macroscopic methods for computing their ground state degeneracy. These distinct but related studies illustrate the power of using physical arguments and mathematical proofs in the simulation, design, and computing aspects of topological phases.