Mathematical Foundations of Quantum Computing for Computer Science Researchers

TL;DR

This paper introduces the mathematical foundations of quantum computing for computer science researchers, focusing on qubit representation, SU(2), SO(3), and their relation to quantum operators.

Contribution

It provides an accessible explanation of the mathematical basis of quantum computation, emphasizing the relationship between SU(2), SO(3), and qubit representations.

Findings

Representation of qubits using the Bloch sphere

Relationship between SU(2) and SO(3)

Mapping rotations in SO(3) to SU(2) matrices

Abstract

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.