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arXiv:1601.05288 · Journal of Physics Conference Series

An Invitation to the Mathematics of Topological Quantum Computation

E C Rowell

Published Mar 1, 2016• 12 citations
arXivHTMLPDF

Abstract

Two-dimensional topological states of matter offer a route to quantum computation that would be topologically protected against the nemesis of the quantum circuit model: decoherence. Research groups in industry, government and academic institutions are pursuing this approach. We give a mathematician's perspective on some of the advantages and challenges of this model, highlighting some recent advances. We then give a short description of how we might extend the theory to three-dimensional materials.

DOI
10.1088/1742-6596/698/1/012012

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Two-dimensional topological states of matter offer a route to quantum computation that would be topologically protected against the nemesis of the quantum circuit model: decoherence. Research groups in industry, government and academic institutions are pursuing this approach. We give a mathematician's perspective on some of the advantages and challenges of this model, highlighting some recent advances. We then give a short description of how we might extend the theory to three-dimensional materials.
Quantum computation represents a paradigm shift in information processing, promising exponential speedups for specific computational tasks. However, a significant hurdle to realizing practical quantum computers lies in the phenomenon of quantum decoherence – the loss of quantum information due to interactions with the surrounding environment. This work explores the potential of topological quantum computation (TQC) as a robust solution to mitigate decoherence. TQC leverages exotic states of matter, specifically surface states of topological insulators and quantum spin Hall systems, to create protected quantum bits (qubits). These states exhibit inherent topological protection, meaning their quantum properties are robust against local perturbations, drastically reducing the detrimental effects of decoherence. We detail the core principles of TQC, focusing on how the topological nature of these materials safeguards quantum information. We will demonstrate how this approach minimizes the influence of environmental noise, leading to improved qubit coherence times and, consequently, enhanced fidelity in quantum computations. This research contributes to the ongoing exploration of TQC as a viable pathway towards scalable and fault-tolerant quantum computing.
Topological quantum computation relies on the concepts of non-Abelian anyons and topological superconductors, holding great promise for noise-protected computation scalable since it stores quantum information coherently, irrespective of position. Included in this review are fundamentals like Majorana fermion zero modes, current innovations at Microsoft regarding nanowires, along with challenges and plans for scaling quantum computation at the level of million-qubits by the year 2030. It employs the use of anyons, a kind of quasi particle that exists in a two-dimensional setting. The world lines of the anyons cross each other in a braid fashion in the three-dimensional spacetime (one time-dimensional and two space-dimensional). The braids correspond to the logic gates in a computer. The one big advantage that the use of the quantum braid has over the trapped quantum particles has to do with stability. A small disturbance in a system may cause the decoherence of the quantum states and errors in calculation in a regular quantum computer, but in the braids, the disturbance does not affect the topological properties because it is like the difference between cutting and reconnecting a string to create a new braid compared to a ball (standing in place for a regular trapped quantum particle in four-dimensional spacetime) hitting a wall. 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. That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.
Quantum entanglement, a cornerstone of quantum mechanics, exhibits unique properties that challenge classical intuition. This paper investigates the potential of quantum entanglement to manifest within non-linear topological structures, exploring novel quantum computation models based on this principle. We propose a framework that leverages the inherent topological properties of quantum states, aiming to enhance entanglement stability and computational efficiency. The core mechanism focuses on optimizing quantum circuit topology to mitigate entanglement decoherence, a persistent challenge in quantum information processing. The research explores the impact of topological structure on quantum entanglement, providing a novel approach to improve the performance of quantum algorithms. This work offers a new perspective on harnessing quantum entanglement for advanced computational applications, potentially revolutionizing the landscape of quantum information theory.
This paper presents a novel theoretical model for quantum decoherence based on topological hidden-variable field theory. The core claim is to establish a model that elucidates the physical mechanism of quantum decoherence and offers a new theoretical foundation for research into quantum computation's resilience against decoherence. We posit that quantum decoherence can be viewed as an emergent phenomenon arising from interactions mediated by hidden variables within a topological hidden-variable field structure. The model utilizes mathematical tools derived from topological hidden-variable field theory to formulate and analyze the underlying physical processes. We demonstrate how the topological nature of the hidden-variable fields influences the decoherence process, providing a fresh perspective on this fundamental challenge in quantum information science. The model aims to bridge the gap between quantum mechanics and topological physics, potentially leading to innovative strategies for mitigating decoherence in quantum computing systems. Key parameters and variables are defined within the model, and the theoretical framework is presented with a focus on providing a conceptual understanding of the decoherence mechanism.
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.