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.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.

Open Journal of Mathematical SciencesApr 23, 20250 citations

THE BASIC THEORY OF SUPERCONDUCTING QUANTUM COMPUTING GATE OPERATION AND ITS FIDELITY MEASUREMENT

Feifan Su, Zhaohua YANG, Yonghe DENG

For the existing superconducting qubit devices, constructing the corresponding high-fidelity gate operation after system calibration is a key step to realize quantum computing. As the fifth chapter of the series of “Introduction to physics frontier—superconducting quantum computing”, this paper systematically discusses the physical picture and basic method of constructing superconducting qubit gate operation through microwave pulse, gives a set of construction process of single bit gate operation, and extends the measurement and optimization development of gate operation fidelity. This paper aims to help the physics teachers, senior undergraduates, graduate students, and science and engineering backgrounds interested in superconducting quantum computing readers to systematically understand the whole basic process of superconducting qubit gate construction and gate operation fidelity measurement and optimization.For the existing superconducting qubit devices, constructing the corresponding high-fidelity gate operation after system calibration is a key step to realize quantum computing. As the fifth chapter of the series of “Introduction to physics frontier—superconducting quantum computing”, this paper systematically discusses the physical picture and basic method of constructing superconducting qubit gate operation through microwave pulse, gives a set of construction process of single bit gate operation, and extends the measurement and optimization development of gate operation fidelity. This paper aims to help the physics teachers, senior undergraduates, graduate students, and science and engineering backgrounds interested in superconducting quantum computing readers to systematically understand the whole basic process of superconducting qubit gate construction and gate operation fidelity measurement and optimization.

Wuli yu gongcheng.Jun 1, 20240 citations

Computation with Qubits

WORLD SCIENTIFIC eBooksJul 15, 20200 citations

Manipulating qubits

Michel Le Bellac

Cambridge University Press eBooksJun 15, 20061 citation

Fundamentals of Quantum Computation

This chapter briefly describes the basic concepts and principles of quantum computing. Firstly, the concepts of qubit, quantum coherence, quantum decoherence, quantum entanglement, quantum density operators, linear operators, inner products, outer products, tensor products, Hermite operators, and unitary operators are described. Then, the four basic assumptions of quantum mechanics are introduced, focusing on the measurement assumptions of quantum mechanics. Finally, the definition of commonly used quantum logic gates is given including single qubit gates, double qubit gates, and multiple qubit gates. These contents provide the necessary theoretical basis for subsequent chapters.This chapter briefly describes the basic concepts and principles of quantum computing. Firstly, the concepts of qubit, quantum coherence, quantum decoherence, quantum entanglement, quantum density operators, linear operators, inner products, outer products, tensor products, Hermite operators, and unitary operators are described. Then, the four basic assumptions of quantum mechanics are introduced, focusing on the measurement assumptions of quantum mechanics. Finally, the definition of commonly used quantum logic gates is given including single qubit gates, double qubit gates, and multiple qubit gates. These contents provide the necessary theoretical basis for subsequent chapters.

Advances in computer and electrical engineering book seriesAug 26, 20200 citations

\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.