Fundamentals of Quantum Computing: Theory and Practice

✍ Scribed by Venkateswaran Kasirajan

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 480
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This introductory book on quantum computing includes an emphasis on the development of algorithms.  Appropriate for both university students as well as software developers interested in programming a quantum computer, this practical approach to modern quantum computing takes the reader through the required background and up to the latest developments.

Beginning with introductory chapters on the required math and quantum mechanics, Fundamentals of Quantum Computing proceeds to describe four leading qubit modalities and explains the core principles of quantum computing in detail. Providing a step-by-step derivation of math and source code, some of the well-known quantum algorithms are explained in simple ways so the reader can try them either on IBM Q or Microsoft QDK. The book also includes a chapter on adiabatic quantum computing and modern concepts such as topological quantum computing and surface codes.

Features:

o   Foundational chapters that build the necessary background on math and quantum mechanics.

o   Examples and illustrations throughout provide a practical approach to quantum programming with end-of-chapter exercises.

o   Detailed treatment on four leading qubit modalities -- trapped-ion, superconducting transmons, topological qubits, and quantum dots -- teaches how qubits work so that readers can understand how quantum computers work under the hood and devise efficient algorithms and error correction codes. Also introduces protected qubits - 0-π qubits, fluxon parity protected qubits, and charge-parity protected qubits.

o   Principles of quantum computing, such as quantum superposition principle, quantum entanglement, quantum teleportation, no-cloning theorem, quantum parallelism, and quantum interference are explained in detail.  

A dedicated chapter on quantum algorithm explores both oracle-based, and Quantum Fourier Transform-based algorithms in detail with step-by-step math and working code that runs on IBM QisKit and Microsoft QDK. Topics on EPR Paradox, Quantum Key Distribution protocols, Density Matrix formalism, and Stabilizer formalism are intriguing. While focusing on the universal gate model of quantum computing, this book also introduces adiabatic quantum computing and quantum annealing.

This book includes a section on fault-tolerant quantum computing to make the discussions complete. The topics on Quantum Error Correction, Surface codes such as Toric code and Planar code, and protected qubits help explain how fault tolerance can be built at the system level.

✦ Table of Contents

Acknowledgments
Contents
Image Credits
Reading Guide
Introduction
The Audience of this Book
Opportunities for the Readers
Structure of the Book
Chapter 1 - Foundations of Quantum Mechanics
Chapter 2 - Dirac´s bra-ket notation and Hermitian Operators
Chapter 3 - The Quantum Superposition Principle and Bloch Sphere Representation
Chapter 4 - Qubit Modalities
Chapter 5 - Quantum Circuits and DiVincenzo Criteria
Chapter 6 - Quantum Communications
Chapter 7 - Quantum Algorithms
Chapter 8 - Adiabatic Optimization and Quantum Annealing
Chapter 9 - Quantum Error Correction
Chapter 10 - Conclusion
Reading Guide
Note:
Note:
Conventions
Indexing Qubits
Coloring Scheme for Bloch Spheres
Source Code Listings
Keeping the Connections
About the Author
Part I - Foundations
1: Foundations of Quantum Mechanics
1.1 Matter
1.1.1 States of Matter
1.2 Atoms, Elementary Particles, and Molecules
1.2.1 Atoms
1.2.2 The Periodic Table
1.2.3 Bohr´s Model of the Atom
1.2.4 Elementary Particles
1.2.5 Molecules
1.3 Light and Quantization of Energy
1.3.1 Blackbody Radiation
1.3.2 Photoelectric Effect and the Concept of Photons
1.3.3 What Happens When Light Hits an Atom?
1.4 Electron Configuration
1.4.1 Pauli´s Exclusion Principle
1.4.2 Principal Quantum Number n
1.4.3 Orbital Angular Momentum Quantum Number l
1.5 Wave-Particle Duality and Probabilistic Nature
1.5.1 Maxwell´s Electromagnetic Theory
1.5.2 De Broglie Waves
1.5.3 Young´s Double-Slit Experiment
1.6 Wavefunctions and Probability Amplitudes
1.6.1 Realism in Physics
1.6.2 What Happens When We Measure a Quantum System?
1.6.3 Heisenberg´s Uncertainty Principle
1.6.4 Weak Measurements
Example 1.1
1.7 Some Exotic States of Matter
1.7.1 Quasiparticles and Collective Excitations
1.7.2 Phonons
1.7.3 Cooper Pairs (or the BCS Pairs)
1.7.4 Superconductors
1.7.5 s-Wave, d-Wave, and p-Wave Superconductors
1.7.6 Anyons
1.7.7 Majorana Fermions
1.8 Summary
Practice Problems
References
2: Dirac´s Bra-ket Notation and Hermitian Operators
2.1 Scalars
2.2 Complex Numbers
2.2.1 Complex Conjugate
2.2.2 Norm of a Complex Number
2.3 Vectors
2.3 Example 2.1
2.3.1 Magnitude of Vectors
2.3.2 Unit Vector
2.3.3 Addition and Subtraction of Vectors
2.3.4 Multiplication of Vectors
2.3.4.1 Dot Product or Scalar Product of Vectors
2.3.4.2 Vector Product or Cross Product of Vectors
2.4 Matrices
2.4.1 Properties of Matrices
2.4 Example 2.2
2.4 Example 2.3
2.4 Example 2.4
2.4 Example 2.5
2.4 Example 2.6
2.4 Exercise 2.1
2.4 Exercise 2.2
2.4 Exercise 2.3
2.4.2 Permutation Matrices
2.4.3 Hermitian Matrix
2.4.4 Unitary Matrices
2.4.5 Pauli Matrices
2.4 Exercise 2.4
2.4 Exercise 2.5
2.5 Linear Vector Spaces
2.5.1 Linear Independence
2.5.2 The Dimensionality of the Vector Space
2.5.2.1 Basis
2.5.2.2 Standard Basis
2.5.2.3 Addition of Two Vectors
2.5.2.4 Multiplication by a Scalar
2.5.3 Inner Product or the Scalar Product
2.5.3.1 Orthogonal Vectors
2.5.3.2 Norm of the Vector
2.5.3.3 Orthonormal Basis
2.5 Example 2.7
2.5 Exercise 2.6
2.6 Using Dirac´s Bra-ket Notation
2.6.1 Operators
2.6.2 Properties of Operators
2.6.2.1 Addition and Subtraction
2.6.2.2 Product of Two Operators
2.6.2.3 Identity Operator
2.6.2.4 Associative Law
2.6.2.5 Commutative Property
2.6.2.6 Anticommutator
2.6 Exercise 2.7
2.6 Example 2.8
2.6.2.7 Power
2.6.3 Linear Operators
2.6 Example 2.9
2.6.3.1 The Inverse of an Operator
2.6.4 Hermitian Operators
2.6 Example 2.10
2.6 Example 2.11
2.6 Example 2.12
2.6.4.1 Anti-Hermitian Operators
2.6.4.2 Unitary Operators
2.6 Exercise 2.8
2.6 Exercise 2.9
2.6.5 Outer Product or the Projection Operators
2.6 Exercise 2.10
2.7 Expectation Values and Variances
2.8 Eigenstates, Eigenvalues, and Eigenfunctions
2.8 Example 2.13
2.8 Example 2.14
2.8 Condition Number:
2.9 Characteristic Polynomial
2.9 Example 2.15
2.10 Definite Symmetric Matrices
2.10.1 Positive Semidefinite
2.10.2 Negative Semidefinite
2.11 Tensors
2.11.1 Direct Sum
2.11 Example 2.16
2.11.2 Tensor Product
2.11 Example 2.17
2.11.2.1 Tensor Product of 2 x 2 Matrices
2.11 Exercise 2.11
2.12 Statistics and Probability
2.12.1 Definitions
2.12 Exercise 2.12
2.12.2 Measures of Location
2.12.3 Measures of Dispersion
2.12.4 Probability
2.12 Exercise 2.13
2.12 Example 2.18
2.12.5 Permutation
2.12 Example 2.19
2.12.6 Combinations
2.12 Exercise 2.14
2.13 Summary
Practice Problems
References
3: The Quantum Superposition Principle and Bloch Sphere Representation
3.1 Euclidian Space
3.2 Metric Space
3.2.1 Completeness and Cauchy Sequences
3.2 Example 3.1
3.3 Hilbert Space
3.4 Schrödinger Equation
3.4.1 Schrödinger´s Cat Thought Experiment (Do Not Try This at Home!)
3.4.1.1 Cat State
3.5 Postulates of Quantum Mechanics
3.5.1 Postulate 1
3.5.2 Postulate 2
3.5.3 Postulate 3
3.5.4 Postulate 4
3.5.5 Postulate 5
3.5 Example 3.2
3.5.6 Symmetric and Antisymmetric Wavefunctions
3.6 Quantum Tunneling
3.6 Note:
3.7 Stern and Gerlach Experiment
3.7.1 Ladder Operators (or Raising/Lowering Operators)
3.7 Exercise 3.1
3.8 Bloch Sphere Representation
3.8 Example 3.3
3.8.1 Qubit Rotations
3.8.1.1 Pauli X-Gate
3.8.1.2 Pauli Y-Gate
3.8.1.3 Pauli Z-Gate
3.8.1.4 Rotation Operator
3.8 Example 3.4
3.9 Projective Measurements
3.9.1 Projection Operators
3.9 Exercise 3.2
3.9.2 Measuring Multi-Qubit Systems
3.9.3 Measurements
3.9.4 Positive Operator Valued Measure (POVM)
3.10 Qudits
3.11 Summary
Practice Problems
References
Part II - Essentials
4: Qubit Modalities
4.1 The Vocabulary of Quantum Computing
4.2 Classical Computers-A Recap
4.2.1 Fundamental Difference
4.3 Qubits and Usability
4.4 Noisy Intermediate Scale Quantum Technology
4.5 Qubit Metrics
4.5.1 Spin or Energy Relaxation Time (T1)
4.5.2 Dephasing Time (T)
4.5.3 Decoherence Time (T2)
4.5.4 Hann Echo (T2)
4.5.5 Ramsay Experiment
4.5.6 Gate Time
4.5.7 Gate Fidelity
4.6 Leading Qubit Modalities
4.6.1 Trapped Ion Qubit
4.6.2 Superconducting Transmons
4.6.2.1 Josephson Junction (JJ)
4.6.2.2 Charge Qubit/Cooper Pair Box
4.6.2.3 Transmon Qubits
4.6.3 Topological Qubits
4.6.3.1 Majorana Fermions
4.6.3.2 Topological Quantum Computing
4.6.3.3 Abelian and Non-Abelian Anyons
4.6.3.4 Fusion Properties
4.6 Exercise 4.1
4.6 Exercise 4.2
4.6.4 Anyon Model
4.6.4.1 Fibonacci Anyons
4.6.4.2 Ising Anyons
4.6.4.3 Sources of Anyons
4.6.4.4 Majorana Qubit
4.6.4.5 Nobel Prize-2016
4.6.5 Electron Spin/Quantum Dot
4.6.5.1 Qubit Physics
4.6.5.2 Qubit Initialization
4.6.5.3 Measurement
4.6.5.4 Gate Operations
4.6.5.5 Summary
4.7 A Note on the Dilution Refrigerator
4.7.1 How Does the Dilution Fridge Cool the Qubits?
4.8 Summary
Practice Problems
References
5: Quantum Circuits and DiVincenzo Criteria
5.1 Setting Up the Development Environment
5.1.1 Setting Up IBM Q
5.1.1.1 Installing Qiskit
5.1.2 Installing Microsoft Quantum Development Kit
5.1.3 Verifying the Installation
5.1.3.1 Verifying Qiskit
5.1.3.2 Verifying Microsoft Quantum Kit
5.1.4 Keeping the Installation Updated
5.2 Learning Quantum Programming Languages
5.2.1 Programming for OpenQASM
5.2.2 QDK Using Q#
5.2.3 Qiskit Python
5.2.4 Comparing the Development Platforms
5.3 Introducing Quantum Circuits
5.3.1 On Quantum Measurements
5.3.2 On the Compute Stage
5.4 Quantum Gates
5.4.1 Clifford Group Gates for Single Qubits
5.4.1.1 The Bit-Flip Gate or the NOT Gate or the Pauli X-Gate
5.4.1.2 The Identity Gate
5.4.1.3 The Hadamard Gate-H-Gate
5.4.1.4 Pauli Y Gate
5.4.1.5 Pauli Z Gate
5.4.1.6 The S Gate or the Z90 Gate or the Phase Gate
5.4.1.7 The S Gate or the Sdag Gate
5.4.2 Arbitrary Rotation of the Qubit
5.4.2.1 RX-Gate
5.4.2.2 RY-Gate
5.4.2.3 RZ-Gate
5.4.2.4 T Gate and the T Gate or the Gate
5.4.3 Physical Gates
5.4.3.1 U3-Gate
5.4.3.2 U2-Gate
5.4.3.3 U1-Gate
5.4.4 Multiqubit Gates
5.4.5 Representing a Multi Qubit State
5.4.5.1 The SWAP Gate
5.4.5.2 Square Root of SWAP Gate
5.4.6 Controlled U Gates
5.4.6.1 The Controlled NOT (CNOT) or the cX Gate
5.4.6.2 cY and cZ (CPHASE) Gates
5.4.6.3 Controlled Hadamard Gate or the cH Gate
5.4.6.4 Controlled RZ-Gate or the cRz Gate
5.4.6.5 Controlled U3-Gate or the cU3 Gate
5.4.6.6 Controlled U1-Gate or the cU1 Gate
5.4.7 Extended Gates
5.4.7.1 The Toffoli or the CCNOT Gate or the ccX Gate
5.4.7.2 The Fredkin or the CSWAP or the cS Gate
5.4.7.3 The Deutsch Gate D(θ)
5.4.8 Universality of Quantum Gates
5.4.8.1 Global Phase and Its Irrelevance
5.4.8.2 Sequence of Gates
5.4.8.3 Universality
5.4.9 Circuit Optimization
5.4.9.1 Hardware Topology
5.4.9.2 Transpilation
5.4.9.3 Circuit Depth, Width, and Size
5.4.9.4 Barrier Gate
5.4.9.5 The Reset Gate
5.4.10 State Visualization
5.4.10.1 Plot_Bloch_Multivector
5.4.10.2 Plot_State_City
5.4.11 Gottesman-Knill Theorem
5.5 The Compute Stage
5.5.1 Experimenting with the CNOT Gate
5.5.2 Implementing Digital Logic Gates Using Quantum Circuits
5.5.2.1 Quantum NOT Gate
5.5.2.2 Quantum OR Gate
5.5.2.3 Quantum AND Gate
5.5.2.4 Quantum XOR Gate
5.5 Exercise 5.1
5.5.2.5 Quantum Full Adder Circuit
5.5 Exercise 5.2
5.5.3 Implementing a CNOT Gate with a gate
5.6 Quantum Entanglement
5.6 Exercise 5.3
5.7 No-Cloning Theorem
5.8 Quantum Teleportation
5.8 Exercise 5.4
5.9 Superdense Coding
5.10 Greenberger-Horne-Zeilinger State (GHZ State)
5.10 Exercise 5.5
5.11 Walsh-Hadamard Transform
5.12 Quantum Interference
5.13 Phase Kickback
5.14 DiVincenzo´s criteria for Quantum Computation
5.14.1 A Scalable Physical System with Well-Characterized Qubit
5.14.2 The Ability to Initialize the State of the Qubits to a Simple Fiducial State
5.14.3 Long Relevant Decoherence Times
5.14.4 A ``Universal´´ Set of Quantum Gates
5.14.5 A Qubit-Specific Measurement Capability
5.14.6 The Ability to Interconvert Stationary and Flying Qubits
5.14.7 The Ability to Faithfully Transmit Flying Qubits between Specified Locations
5.15 Summary
Practice Problems
References
6: Quantum Communications
6.1 EPR Paradox
6.1.1 Bell Measurement
6.1 Exercise 6.1
6.1.2 Local Realism
6.1.3 Bell´s Inequalities
6.2 Density Matrix Formalism
6.2.1 Maximally Mixed States
6.2.2 Unitary Transformations on Density Matrices
6.2.3 Eigen Decomposition
6.2.4 Verification of Entanglement
6.2.5 Reduced Density Matrices
6.2.6 No-Communication Theorem
6.2 Exercise 6.2
6.3 Von Neumann Entropy
6.3.1 Schumacher´s Compression Theorem
6.3.2 Bloch Sphere
6.3.3 Schmidt Decomposition and Degree of Entanglement
6.3.4 Purification
6.3 Example 6.1
6.4 Photons
6.4.1 Polarization
6.4.2 Entangled Photons
6.4.2.1 Cascade Experiments
6.4.2.2 Spontaneous Parametric Down Conversion
6.5 Quantum Communication
6.6 The Quantum Channel
6.7 Quantum Communication Protocols
6.7.1 BB84 Protocol
6.7.2 Ekert 91 Protocol
6.8 RSA Security
6.8.1 Modular Exponentiation
6.8.2 Period Estimation
6.9 Summary
Practice Problems
References
7: Quantum Algorithms
7.1 Quantum Ripple Adder Circuit
7.1 Exercise 7.1
7.2 Quantum Fourier Transformation
7.2.1 Classical Fourier Transformation
7.2.2 Discrete Fourier Transform (DFT)
7.2 Example 7.1
7.2.3 Quantum Fourier Transformation
7.2.4 Testing the QFT
7.2 Exercise 7.2
7.3 Deutsch-Jozsa Oracle
7.3.1 Deutsch Oracle
7.3.2 Deutsch-Jozsa Oracle
7.3 Exercise 7.3
7.4 The Bernstein-Vazirani Oracle
7.5 Simon´s Algorithm
7.6 Quantum Arithmetic Using QFT
7.6.1 Simple QFT Adder
7.6.2 Controlled QFT Adder
7.6.3 QFT-Based Multiplier
7.6 Exercise 7.4
7.7 Modular Exponentiation
7.7 Exercise 7.5
7.8 Grover´s Search Algorithm
7.8 Exercise 7.6
7.9 Shor´s Algorithm
7.9.1 Factoring Number 15
7.9.2 Quantum Circuit to Factorize 15
7.10 A Quantum Algorithm for K-Means
7.10.1 Quantum Distance Estimation
7.11 Quantum Phase Estimation (QPE)
7.12 HHL Algorithm for Solving Linear Equations
7.13 Quantum Complexity Theory
7.14 Summary
Practice Problems
References
8: Adiabatic Optimization and Quantum Annealing
8.1 Adiabatic Evolution
8.2 Proof of the Adiabatic Theorem
8.2.1 Berry Phase
8.3 Adiabatic Optimization
8.4 Quantum Annealing
8.5 Summary
Practice Problems
References
9: Quantum Error Correction
9.1 Classical Error Correction
9.1.1 Shannon´s Entropy
9.1.2 Communication on a Classical Channel
9.1.3 Classical Error Correction Codes
9.1 Example 9.1
9.2 Quantum Error Codes
9.2.1 Quantum Error Correction
9.2.2 Quantum Code
9.2.3 Pauli Group
9.2 Exercise 9.1
9.2.4 Correcting Bit-Flips
9.2.5 Correcting Phase-Flips
9.3 Stabilizer Formalism
9.3.1 Gottesman-Knill Theorem
9.3.2 Shor´s 9-Qubit Error Correction Code
9.3.3 CSS Codes
9.3.4 Steane´s 7-Qubit Error Correction Code
9.3 Exercise 9.2
9.4 The Path Forward: Fault-Tolerant Quantum Computing
9.4.1 Performing a Fault-Tolerant Procedure
9.4.2 Performing Error Corrected Measurements
9.4.3 Performing Robust Quantum Computation
9.5 Surface Codes
9.5.1 Toric Code
9.5.1.1 Summary
9.5.2 Planar Code (or Surface Code)
9.5.2.1 Logical Qubit
9.5.2.2 Error Handling
9.5.3 Summary
9.6 Protected Qubits
9.6.1 0-π Qubit
9.6.1.1 Qubit Physics
9.6.1.2 Measurement
9.6.1.3 Protected Phase Gate
9.6.2 Fluxon-Parity Protected Superconducting Qubit
9.6.2.1 Qubit Physics
9.6.2.2 Protection
9.6.3 Parity Protected Superconductor-Semiconductor Qubit
9.6.3.1 Qubit Design
9.6.3.2 Protection
9.6.4 Summary
Practice Problems
References
10: Conclusion
10.1 How Many Qubits Do We Need?
10.2 Classical Simulation
10.3 Backends Today
10.4 Future State
References
Appendix
Appendix
Index