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Mathematics of Quantum Computing: An Introduction

Wolfgang Scherer

Published Nov 13, 2019• 7 citations
Publisher
DOI
10.1007/978-3-030-12358-1

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This paper provides a short introduction to the mathematical foundation of quantum computation for researchers in computer science by providing an introduction fo the mathematical basis of calculations. This paper concerns the mathematical foundations of quantum computation addressing first the representation of qubit using the Bloch sphere and second the special relations between SU(2) and SO(3). The properties of SU(2) are introduced focusing especially about the double-covering of SO(3) and explaining how to map rotations of SO(3) into matrices of SU(2). Quantum physic operators are based on SU(2) since we have a direct relationship to SO(3) namely one isomorphism. We start first from basic representations of qubit in R^3 and representations of operators in SU(2) and we next discuss with operators that permit to move from one SU(2) to another one according to a specific operator of SU(2) that is related to rotation into R^3.
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Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction to some of the ideas in quantum computing. The paper begins by motivating the central ideas of quantum mechanics and quantum computation with simple toy models. From there we move on to a formal presentation of the small fraction of (finite dimensional) quantum mechanics that we will need for basic quantum computation. Central notions of quantum architecture (qubits and quantum gates) are described. The paper ends with a presentation of one of the simplest quantum algorithms: Deutsch's algorithm. Our presentation demands neither advanced mathematics nor advanced physics.