Year: 2025
Type: article
Abstract: We present an introduction to the mathematics of quantum physics and quantum computation which put emphasis on the basic mathematical aspects of definition and operations on qubits. We start by a comprehensive introduction of a qubit as a unit element of \( \mathbb{C}^2 \), and its representations on spheres in \( \mathbb{R}^3 \). This introduction leads to the interpretation of Pauli operators as basic rotations in \( \mathbb{R}^3 \). Then we study unitary operators. Their link to rotations in \( \mathbb{R}^3 \) is established using the density operator associated to a qubit. We complete this paper by some decomposition, or splitting, problems of unitary operators on \( \mathbb{C}^2 \) based on decomposition results of rotations in \( \mathbb{R}^3 \). These decomposition results are useful for the construction of quantum gates.
Authors: Département de mathématiques Université de Sherbrooke Sherbrooke (Qc), Canada, J1K 2R1, François Dubeau
Institution: Université de Sherbrooke
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4409723110