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CRC Press eBooks

Nuts and Bolts of Quantum Mathematics

M. S. Ramkarthik, Payal D. Solanki

Published Jun 29, 2021• 0 citations
Publisher

Abstract

This chapter contains the recipes for the operations which will be used in manipulating quantum states and operators. It contains recipes to perform normalization, inner and outer products, tensor products, commutators and anti-commutators, inter-conversion between binary and decimal, bit testing, swapping and shifts of qubits. These are the most important and fundamental operations involving quantum states and operators.

DOI
10.1201/9781003164678-2

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1 Preface 2 Introduction 2.1 Computing and the Laws of Physics 2.2 Quantum Information 2.3 Quantum Computers 2.4 The Wave and the Corpuscular Nature of Light 2.5 Deterministic versus Probabilistic Photon Behavior 2.6 State Description, Superposition, and Uncertainty 2.7 Measurements in Multiple Bases 2.8 Measurements of Superposition States 2.9 An Augmented Probabilistic Model. The Superposition Probability Rule. 2.10 A Photon Coincidence Experiment 2.11 A Three Beam Splitter Experiment 2.12 BB84, the Emergence of Quantum Cryptography 2.13 A Qubit of History 2.14 Summary and Further Readings 2.15Exercises and Problems 3 Quantum Mechanics, a Mathematical Model of the Physical World 3.1 Vector Spaces 3.2 n-Dimensional Real Euclidean Vector Space 3.3 Linear Operators and Matrices 3.4 Hermitian Operators in a Complex n -Dimensional Euclidean Vector Space 3.5 n -Dimensional Hilbert Spaces. Dirac Notations 3.6 The Inner Product in an n -Dimensional Hilbert Space 3.7 Tensor and Outer Products 3.8 Quantum States 3.9 Quantum Observables. Quantum Operators 3.10 Spectral Decomposition of a Quantum Operator 3.11 The Measurement of Observables 3.12 More about Measurements. The Density Operator 3.13 Double-Slit Experiments 3.14 Stern-Gerlach Type Experiments 3.15 The Spin as an Intrinsic Property 3.16 SchrodingerOs Wave Equation 3.17 HeisenbergOs Uncertainty Principle 3.18 A Brief History of Quantum Ideas 3.19 Summary and Further Readings 3.20 Exercises and Problems 4 Qubits and Their Physical Realization 4.1 One Qubit, a Very Small Bit 4.2 The Bloch Sphere Representation of One Qubit 4.3 Rotation Operations on the Bloch Sphere 4.4 The Measurement of a Single Qubit 4.5 Pure and Impure States of a Qubit 4.6 A Pair of Qubits. 4.7 The Fragility of Quantum Information. SchrodingerOs Cat 4.8 Qubits: from Hilbert Spaces to Physical Implementation 4.9 Qubits as Spin One-Half Particles 4.10 The Measurement of the Spin 4.11 The Qubit as a Polarized Photon 4.12 4.13 The Exchange of Information Using Entangled Particles 4.14 Summary and Further Readings 4.15 Exercises and Problems 5 Quantum Gates and Quantum Circuits 5.1 Classical Logic Gates and Circuits 5.2 One-Qubit Gates 5.3 The Hadamard Gate, Beam Splitters and Interferometers 5.4 Two-Qubit Gates. The CNOT Gate 5.5 Can We Build Quantum Copy Machines? 5.6 Three-Qubit Gates. The Fredkin Gate 5.7 The Toffoli Gate 5.8 Quantum Circuits 5.9 The No Cloning Theorem 5.10 Qubit Swapping and Full Adder Circuits 5.11 More about Unitary Operations and Rotation Matrices 5.12 Single-Qubit Controlled Operations 5.13 Multiple Qubit Controlled Operations 5.14 Universal Quantum Gates 5.15 A Quantum Circuit for the Walsh-Hadamard Transform 5.16 The State Transformation Performed by Quantum Circuits 5.17 Mathematical Models of a Quantum Computer 5.18 Errors, Uniformity Conditions, and Time Complexity 5.19 Summary and Further Readings 5.20 Exercises and Problems 6 Quantum Algorithms 6.1 From Classical to Quantum Turing Machines 6.2 Computational Complexity and 6.3 Classes of Quantum Algorithms 6.4 Quantum Parallelism 6.5 DeutschOs Problem 6.6 Quantum Fourier Transform 6.7 Tensor Product Factorization 6.8 A Circuit for Quantum Fourier Transform 6.9 A Case Study: A Three-Qubit QFT 6.10 ShorOs Factoring Algorithm and Order Finding 6.11 A Quantum Circuit for Computing f(x)Modulo 2 6.12 SimonOs Algorithm for Phase Estimation 6.13 The Fourier Transform on an Abelian Group 6.14 Periodicity and the Quantum Fourier Transform 6.15 The Discrete Logarithms Evaluation Problem 6.16 The Hidden Subgroup Problem 6.17 Quantum Search Algorithms 6.18 Historical Notes 6.19 Summary and Further Readings 6.20 Exercises and Problems 7 The Entanglement of Computing and Communication with Quantum Mechanics. Reversible Computations 7.1 Communication, Entropy, and Quantum Information 7.2 Information Encoding 7.3 Quantum Teleportation with Maximally Entangled Particles 7.4 Anti-Correlation and Teleportation 7.5 Dense Coding 7.6 Quantum Key Distribution 7.7 EPR Pairs and Bell States 7.8 Uncertainty and Locality 7.9 Possible Explanations of the EPR Experiment 7.10 The Bell Inequality. Local Realism 7.11 Reversibility and Entropy 7.12 Thermodynamics and Thermodynamic Entropy 7.13 The Maxwell Demon 7.14 Energy Consumption. Landauer Principle 7.15 Low Power Computing. Adiabatic Switching 7.16 Bennett Information Driven Engine 7.17 Logically Reversible Turing Machines and Physical Reversibility 7.18 Historical Notes 7.19 Summary and Further Readings 297 7.20 Exercises and Problems 299 8 Appendix I: Algebraic Structures 8.1 Rings, Commutative Rings, Integral Domains, Fields 8.2 Complex Numbers 8.3 Abstract Groups and Isomorphisms 8.4 Matrix Representation 8.5 Groups of Transformations 8.6 Symmetry in a Plane 8.7 Finite Fields 9 Appendix II: Modular Arithmetic 9.1 Elementary Number Theory Concepts 9.2 EuclidOs Algorithm for Integers 9.3 The Chinese Remainder Theorem and its Applications 9.4 Computer Arithmetic for Large Integers 10 Appendix III: Welsh-Hadamard Transform 10.1 Hadamard Matrices 10.2 The Fast Hadamard Transform 11 Appendix IV: Fourier Transform and Fourier Series 12 Glossary

Prentice Hall eBooks•Jan 1, 2005•58 citations
This chapter contains the recipes for the operations which will be used in manipulating quantum states and operators. It contains recipes to perform normalization, inner and outer products, tensor products, commutators and anti-commutators, inter-conversion between binary and decimal, bit testing, swapping and shifts of qubits. These are the most important and fundamental operations involving quantum states and operators.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.
This chapter discusses the main operation of tensor product, which is behind almost all quantum information manipulation. Tensor products are exploited in a similar manner to yield quantum operators to be used with quantum states of two or more quantum bits. The magnitude of the inversion depends on the state of the control q-bit. It can be said that the CNOT operator is a NOT operator dependent on a second q-bit. It can be constructed out of two chained H operators. The Toffoli operator can be seen as an extension of the CNOT port. It is considered universal and it is possible to build based on any other quantum operator. The chapter shows that if not handled efficiently, it would require a lot of memory space, which makes the design emulate a quantum processor quite expensive to produce.
This chapter contains recipes of all the major quantum gates used in performing various quantum information tasks. It also has some useful recipes on the construction of N qubit computational basis, N qubit Greenberger-Horne-Zeilinger states, N qubit W states, two-qubit Bell states, N qubit generalized Werner states. Apart from that, entropy and its quantification by various measures like Shannon entropy, linear and Relative entropy are given. It becomes very important in quantum information theory to know the closeness between two quantum states and to this end, recipes related to trace distance, fidelity, super fidelity, Bures distance. The chapter ends with two important recipes which are for computing the expectation values of observables and the quantum mechanical single qubit measurements.
1 Preface 2 Introduction 2.1 Computing and the Laws of Physics 2.2 Quantum Information 2.3 Quantum Computers 2.4 The Wave and the Corpuscular Nature of Light 2.5 Deterministic versus Probabilistic Photon Behavior 2.6 State Description, Superposition, and Uncertainty 2.7 Measurements in Multiple Bases 2.8 Measurements of Superposition States 2.9 An Augmented Probabilistic Model. The Superposition Probability Rule. 2.10 A Photon Coincidence Experiment 2.11 A Three Beam Splitter Experiment 2.12 BB84, the Emergence of Quantum Cryptography 2.13 A Qubit of History 2.14 Summary and Further Readings 2.15Exercises and Problems 3 Quantum Mechanics, a Mathematical Model of the Physical World 3.1 Vector Spaces 3.2 n-Dimensional Real Euclidean Vector Space 3.3 Linear Operators and Matrices 3.4 Hermitian Operators in a Complex n -Dimensional Euclidean Vector Space 3.5 n -Dimensional Hilbert Spaces. Dirac Notations 3.6 The Inner Product in an n -Dimensional Hilbert Space 3.7 Tensor and Outer Products 3.8 Quantum States 3.9 Quantum Observables. Quantum Operators 3.10 Spectral Decomposition of a Quantum Operator 3.11 The Measurement of Observables 3.12 More about Measurements. The Density Operator 3.13 Double-Slit Experiments 3.14 Stern-Gerlach Type Experiments 3.15 The Spin as an Intrinsic Property 3.16 SchrodingerOs Wave Equation 3.17 HeisenbergOs Uncertainty Principle 3.18 A Brief History of Quantum Ideas 3.19 Summary and Further Readings 3.20 Exercises and Problems 4 Qubits and Their Physical Realization 4.1 One Qubit, a Very Small Bit 4.2 The Bloch Sphere Representation of One Qubit 4.3 Rotation Operations on the Bloch Sphere 4.4 The Measurement of a Single Qubit 4.5 Pure and Impure States of a Qubit 4.6 A Pair of Qubits. 4.7 The Fragility of Quantum Information. SchrodingerOs Cat 4.8 Qubits: from Hilbert Spaces to Physical Implementation 4.9 Qubits as Spin One-Half Particles 4.10 The Measurement of the Spin 4.11 The Qubit as a Polarized Photon 4.12 4.13 The Exchange of Information Using Entangled Particles 4.14 Summary and Further Readings 4.15 Exercises and Problems 5 Quantum Gates and Quantum Circuits 5.1 Classical Logic Gates and Circuits 5.2 One-Qubit Gates 5.3 The Hadamard Gate, Beam Splitters and Interferometers 5.4 Two-Qubit Gates. The CNOT Gate 5.5 Can We Build Quantum Copy Machines? 5.6 Three-Qubit Gates. The Fredkin Gate 5.7 The Toffoli Gate 5.8 Quantum Circuits 5.9 The No Cloning Theorem 5.10 Qubit Swapping and Full Adder Circuits 5.11 More about Unitary Operations and Rotation Matrices 5.12 Single-Qubit Controlled Operations 5.13 Multiple Qubit Controlled Operations 5.14 Universal Quantum Gates 5.15 A Quantum Circuit for the Walsh-Hadamard Transform 5.16 The State Transformation Performed by Quantum Circuits 5.17 Mathematical Models of a Quantum Computer 5.18 Errors, Uniformity Conditions, and Time Complexity 5.19 Summary and Further Readings 5.20 Exercises and Problems 6 Quantum Algorithms 6.1 From Classical to Quantum Turing Machines 6.2 Computational Complexity and 6.3 Classes of Quantum Algorithms 6.4 Quantum Parallelism 6.5 DeutschOs Problem 6.6 Quantum Fourier Transform 6.7 Tensor Product Factorization 6.8 A Circuit for Quantum Fourier Transform 6.9 A Case Study: A Three-Qubit QFT 6.10 ShorOs Factoring Algorithm and Order Finding 6.11 A Quantum Circuit for Computing f(x)Modulo 2 6.12 SimonOs Algorithm for Phase Estimation 6.13 The Fourier Transform on an Abelian Group 6.14 Periodicity and the Quantum Fourier Transform 6.15 The Discrete Logarithms Evaluation Problem 6.16 The Hidden Subgroup Problem 6.17 Quantum Search Algorithms 6.18 Historical Notes 6.19 Summary and Further Readings 6.20 Exercises and Problems 7 The Entanglement of Computing and Communication with Quantum Mechanics. Reversible Computations 7.1 Communication, Entropy, and Quantum Information 7.2 Information Encoding 7.3 Quantum Teleportation with Maximally Entangled Particles 7.4 Anti-Correlation and Teleportation 7.5 Dense Coding 7.6 Quantum Key Distribution 7.7 EPR Pairs and Bell States 7.8 Uncertainty and Locality 7.9 Possible Explanations of the EPR Experiment 7.10 The Bell Inequality. Local Realism 7.11 Reversibility and Entropy 7.12 Thermodynamics and Thermodynamic Entropy 7.13 The Maxwell Demon 7.14 Energy Consumption. Landauer Principle 7.15 Low Power Computing. Adiabatic Switching 7.16 Bennett Information Driven Engine 7.17 Logically Reversible Turing Machines and Physical Reversibility 7.18 Historical Notes 7.19 Summary and Further Readings 297 7.20 Exercises and Problems 299 8 Appendix I: Algebraic Structures 8.1 Rings, Commutative Rings, Integral Domains, Fields 8.2 Complex Numbers 8.3 Abstract Groups and Isomorphisms 8.4 Matrix Representation 8.5 Groups of Transformations 8.6 Symmetry in a Plane 8.7 Finite Fields 9 Appendix II: Modular Arithmetic 9.1 Elementary Number Theory Concepts 9.2 EuclidOs Algorithm for Integers 9.3 The Chinese Remainder Theorem and its Applications 9.4 Computer Arithmetic for Large Integers 10 Appendix III: Welsh-Hadamard Transform 10.1 Hadamard Matrices 10.2 The Fast Hadamard Transform 11 Appendix IV: Fourier Transform and Fourier Series 12 Glossary