Quantum Computing ─ From Linear Algebra To Physical Realizations

Covering both theory and progressive experiments, Quantum Computing: From Linear Algebra to Physical Realizations explains how and why superposition and entanglement provide the enormous computational power in quantum computing. This self-contained, classroom-tested book is divided into two sections, with the first devoted to the theoretical aspects of quantum computing and the second focused on several candidates of a working quantum computer, evaluating them according to the DiVincenzo criteria.
Topics in Part I

Linear algebra
Principles of quantum mechanics
Qubit and the first application of quantum information processing—quantum key distribution
Quantum gates
Simple yet elucidating examples of quantum algorithms
Quantum circuits that implement integral transforms
Practical quantum algorithms, including Grover’s database search algorithm and Shor’s factorization algorithm
The disturbing issue of decoherence
Important examples of quantum error-correcting codes (QECC)
Topics in Part II

DiVincenzo criteria, which are the standards a physical system must satisfy to be a candidate as a working quantum computer
Liquid state NMR, one of the well-understood physical systems
Ionic and atomic qubits
Several types of Josephson junction qubits
The quantum dots realization of qubits
Looking at the ways in which quantum computing can become reality, this book delves into enough theoretical background and experimental research to support a thorough understanding of this promising field.

From linear algebra to quantum computing Basics of Vectors and Matrices Vector Spaces Linear Dependence and Independence of Vectors Dual Vector Spaces Basis, Projection Operator, and Completeness Relation Linear Operators and Matrices Eigenvalue Problems Pauli Matrices Spectral Decomposition Singular Value Decomposition (SVD) Tensor Product (Kronecker Product) Framework of Quantum Mechanics Fundamental Postulates Some Examples Multipartite System, Tensor Product, and Entangled State Mixed States and Density Matrices Qubits and Quantum Key Distribution Qubits Quantum Key Distribution (BB84 Protocol) Quantum Gates, Quantum Circuit, and Quantum Computer Introduction Quantum Gates Correspondence with Classical Logic Gates No-Cloning Theorem Dense Coding and Quantum Teleportation Universal Quantum Gates Quantum Parallelism and Entanglement Simple Quantum Algorithms Deutsch Algorithm Deutsch–Jozsa Algorithm and Bernstein–Vazirani Algorithm Simon’s Algorithm Quantum Integral Transforms Quantum Integral Transforms Quantum Fourier Transform (QFT) Application of QFT: Period-Finding Implementation of QFT Walsh–Hadamard Transform Selective Phase Rotation Transform Grover’s Search Algorithm Searching for a Single File Searching for d Files Shor’s Factorization Algorithm The RSA Cryptosystem Factorization Algorithm Quantum Part of Shor’s Algorithm Probability Distribution Continued Fractions and Order-Finding Modular Exponential Function Decoherence Open Quantum System Measurements as Quantum Operations Examples Lindblad Equation Quantum Error-Correcting Codes (QECC) Introduction 3-Qubit Bit-Flip Code and Phase-Flip Code Shor’s 9-Qubit Code Calderbank–Shor–Steane (CSS) 7-Qubit QECC DiVincenzo–Shor 5-Qubit QECC Physical realizations of quantum computing DiVincenzo Criteria Introduction DiVincenzo Criteria Physical Realizations Beyond DiVincenzo Criteria NMR Quantum Computer Introduction NMR Spectrometer Hamiltonian Implementation of Gates and Algorithms Time-Optimal Control of NMR Quantum Computer Measurements Preparation of Pseudopure State DiVincenzo Criteria Trapped Ions Introduction Electronic States of Ion as Qubit Ions in Paul Trap Ion Qubit Quantum Gates Readout DiVincenzo Criteria Quantum Computing with Neutral Atoms Introduction Trapping Neutral Atoms 1-Qubit Gate Quantum State Engineering of Neutral Atoms Preparation of Entangled Neutral Atoms DiVincenzo Criteria Josephson Junction Qubits Introduction Nanoscale Josephson Junctions and SQUIDs Charge Qubit Flux Qubit Quantronium Current-Biased Qubit Readout Coupled Qubits DiVincenzo Criteria Quantum Computing with Quantum Dots Introduction Mesoscopic Semiconductors Electron Charge Qubit Electron Spin Qubit DiVincenzo Criteria Appendix: Solutions to Selected Exercises Index