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Journal of Knot Theory and Its Ramifications

A short survey of quantum computing

Uwe Kaiser

Published Dec 22, 2016• 1 citation
Publisher

Abstract

Quantum computing is closely related to [Formula: see text]-dimensional topological quantum field theory through its approach in topological quantum computation. The following are notes based on lectures given by the author at the Mathematics Department of George Washington University. The idea is to give a brief introduction and survey of a few basic facts of quantum computing to an audience of non-experts interested in taking a quick look at some of the most important concepts of quantum computing. The author is a topologist with strong interest in the general ideas of quantum computing. The notes touch on how quantum topology enters the field but emphasizes the general ideas of quantum computing in the first place.

DOI
10.1142/s0218216517410048

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Quantum computing is closely related to [Formula: see text]-dimensional topological quantum field theory through its approach in topological quantum computation. The following are notes based on lectures given by the author at the Mathematics Department of George Washington University. The idea is to give a brief introduction and survey of a few basic facts of quantum computing to an audience of non-experts interested in taking a quick look at some of the most important concepts of quantum computing. The author is a topologist with strong interest in the general ideas of quantum computing. The notes touch on how quantum topology enters the field but emphasizes the general ideas of quantum computing in the first place.
Quantum computing is a field of great interest, attracting, among others, the attention of many mathematicians. Although not all quantum mechanics is needed to successfully engage in research on quantum computing, the somewhat superficial approach usually applied by non-physicists is, in the opinion of the author of the lectures, not feasible. The following notes from lectures given at the mathematics department of George Washington University are meant to be a partial remedy to the situation, offering a very brief and slightly unorthodox introduction to one-particle quantum mechanics, and even shorter discussion of passage to multi-particle quantum mechanics, as needed for quantum computing.
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.
Quantum computing concepts are described using geometric algebra, without using complex numbers or matrices. This novel approach was developed in the first author’s Ph.D. dissertation in Electrical Engineering at University of Texas at Dallas (May 2002) and enables the expression of the principle ideas of quantum computation without requiring an advanced degree in mathematics or physics. Using a topologically derived algebraic notation that relies only on addition and the anticommutative geometric product, this paper describes the following quantum computing concepts: bits, vectors, states, orthogonality, qubits, classical states, superposition states, spinor, reversibility, unitary operator, singular, entanglement, ebits, separability, information erasure, destructive interference and measurement. These central quantum concepts can be described simply in geometric algebra, thereby facilitating the understanding of quantum computing concepts by non-physicists and non-mathematicians.