Preface FOUNDATIONS OF QUANTUM INFORMATICS Spins: The Stern-Gerlach experiment and spin filter EPR, Bell's inequalities, and hidden variables The Landauer principle QUANTUM COMPUTATION AND QUANTUM SYSTEMS Turing machines and binary logic gates Quantum mechanical systems Hilbert spaces Complex finite dimensional Hilbert Spaces Quantum Turing machines Universality of elementary quantum gates Quantum algorithms Quantum adder and multiplier Quantum error correction codes Lasers: a heuristic introduction Quantum computing devices and requirements TWO-LEVEL ATOMS AND CAVITY QED Two-level atoms Quantization of the electromagnetic field Cavity QED Cavity QED for the quantum phase gate Quantum eraser Quantum disentanglement eraser IMPERFECT QUANTUM OPERATIONS Fidelity Density matrices Time evolution of density matrices Examples of master equations Fidelity calculations ION TRAPS Introduction Ion qubits Summary of ion preparation Coherence Quantum gates Large scale confined-ion quantum computer Trap architecture and performance Teleportation of coherent information Experimental DFS logic gates Quantum error correction by ion traps Summary of ion quantum computation QUANTUM LOGIC USING COLD, CONFINED ATOMS Introduction Atom trapping and detection Atom interactions with external fields Atom trapping Qubits and gates Controlled two-qubit gates Coherence properties of atom gates Assessment QUANTUM DOTS QUANTUM COMPUTING GATES Introduction Electrons in quantum dots microcavity Coupled electron spins Biexciton in a single quantum dot Conclusions LINEAR OPTICS COMPUTERS Classical electrodynamics - Classical computers Quantum electrodynamics - Quantum computers Teleportation Summary and outlook SUPERCONDUCTING QUANTUM COMPUTING DEVICES Introduction Superconductivity More on Cooper pairs and Josephson junctions Superconducting circuits: classical Superconducting circuits: quantum Quantum gates Measurement NMR QUANTUM COMPUTING Nuclear magnetic resonance Basic technology with NMR Solid state NMR Shor's algorithm and its experimental realization Quantum algorithm for lattice-gas systems Conclusion Appendix A: The Fock-Darwin States Appendix B: Evaluation of the exchange energy Appendix C: Transformation of quantum states: SU(2) and SO(3) Appendix D: The Homeomorphism from SU(2) to SO(3)
Preface FOUNDATIONS OF QUANTUM INFORMATICS Spins: The Stern-Gerlach experiment and spin filter EPR, Bell's inequalities, and hidden variables The Landauer principle QUANTUM COMPUTATION AND QUANTUM SYSTEMS Turing machines and binary logic gates Quantum mechanical systems Hilbert spaces Complex finite dimensional Hilbert Spaces Quantum Turing machines Universality of elementary quantum gates Quantum algorithms Quantum adder and multiplier Quantum error correction codes Lasers: a heuristic introduction Quantum computing devices and requirements TWO-LEVEL ATOMS AND CAVITY QED Two-level atoms Quantization of the electromagnetic field Cavity QED Cavity QED for the quantum phase gate Quantum eraser Quantum disentanglement eraser IMPERFECT QUANTUM OPERATIONS Fidelity Density matrices Time evolution of density matrices Examples of master equations Fidelity calculations ION TRAPS Introduction Ion qubits Summary of ion preparation Coherence Quantum gates Large scale confined-ion quantum computer Trap architecture and performance Teleportation of coherent information Experimental DFS logic gates Quantum error correction by ion traps Summary of ion quantum computation QUANTUM LOGIC USING COLD, CONFINED ATOMS Introduction Atom trapping and detection Atom interactions with external fields Atom trapping Qubits and gates Controlled two-qubit gates Coherence properties of atom gates Assessment QUANTUM DOTS QUANTUM COMPUTING GATES Introduction Electrons in quantum dots microcavity Coupled electron spins Biexciton in a single quantum dot Conclusions LINEAR OPTICS COMPUTERS Classical electrodynamics - Classical computers Quantum electrodynamics - Quantum computers Teleportation Summary and outlook SUPERCONDUCTING QUANTUM COMPUTING DEVICES Introduction Superconductivity More on Cooper pairs and Josephson junctions Superconducting circuits: classical Superconducting circuits: quantum Quantum gates Measurement NMR QUANTUM COMPUTING Nuclear magnetic resonance Basic technology with NMR Solid state NMR Shor's algorithm and its experimental realization Quantum algorithm for lattice-gas systems Conclusion Appendix A: The Fock-Darwin States Appendix B: Evaluation of the exchange energy Appendix C: Transformation of quantum states: SU(2) and SO(3) Appendix D: The Homeomorphism from SU(2) to SO(3)
1 Making Computation Faster and Communication Secure: Quantum Solution 1.1 Turing Machine: a Real Machine or ... 1.2 ... a Mathematical Procedure 1.3 Faster SuperTuring Computation 1.4 Digital Computers Do not Run on Logic 1.5 Speeding up Computation: Classical Analog Computation ... 1.6 ... versus Quantum Physical Computation 1.7 Complexity Limits: Exponential Time 1.8 Energy Limits ... 1.9 ... and Reversible Gates 1.10 Ultimate Efficiency: Quantum Computers and Qubits 1.11 Combining andMeasuringQubits: QuantumSuperposition?Qubit Primer 1.15 Generating Qubits: Sources of Photons?Polarization Primer 1.16 Correlating Unpolarized Qubits: Quantum Entanglement 1.17 Separating and Transforming Entanglements: Bell States at a Beam Splitter 1.18 Manipulating and Verifying Entanglements: Superdense Coding 1.19 Copying Qubits? No. Teleporting Qubits! 1.20 Unperformed Measurements Have no Values: KochenSpecker Sets 1.21 Controlling Qubits: Quantum Gates and Circuits 1.22 Self-Sustaining Qubits: Quantum Error Correction 1.23 Flying Qubits Connecting Quantum Chips and Computers: Quantum Repeaters 1.24 Why Classical Cryptography Cannot Keep Secrets for Long ... 1.25 ... and why Quantum Cryptography Can? 1.25.1 Entanglement in Action: Deterministic Communication 1.25.2 NoCloning in Action: Probabilistic BB84 Protocol 1.26 WhyThereCanBe noQuantumEavesdroppers? Unconditional Security 2 Quantum Computation and Communication Hardware 2.1 Technological Candidates for Quantum Computation Implementation 2.2 All-Optical Scalable Quantum Computation 2.2.1 Probabilistic Parity Check Gate with a Quantum Encoder 2.2.2 Destructive CNOT Gate 2.2.3 Nondestructive Probabilistic Full CNOT Gate 2.2.4 The Teleportation Trick 2.2.5 Scalable Computation 2.3 Trapped Ions 2.4 Nuclear Magnetic Resonance 2.5 SiliconBased Nuclear Spins?Kane?s Computer 2.6 Quantum Dots 2.6.1 EnergyLevel Design Quantum Dots in a Microcavity 2.6.2 Atom-Cavity Interaction between Levels |0> and |1> 2.6.3 Laser-Atom Interaction between |0> and |1> 2.6.4 Atom-Cavity and Laser-Atom Interactions between |0> and |2> 2.6.5 Implementation of a CNOT gate 2.6.6 Reading out the Results Perspectives 2.6.7 Spinlevel design array of coupled quantum dots 2.7 Superconducting Devices 2.7.1 Josephson Junction 2.7.2 Josephson Junction Circuit and Hamiltonian 2.7.3 Josephson Junction Cooper-Pair Box 2.7.4 Superconducting Quantum Interference Device?SQUID 2.7.5 Cooper Pairs 2.7.6 Quantum Gates for Single Qubits 2.7.7 Quantum Gates for Two Qubits?CNOT Gate 2.7.8 Reading out the results 3 Melting Theory with Implementation 3.1 Quantum Network 3.1.1 One-Atom Laser and Atom-Cavity Coupling 3.1.2 Single Photons on Demand 3.1.3 Dark States 3.1.4 Quantum Repeaters Revisited 3.2 Quantum-Classical Coupling: Interaction-Free Computation 3.3 Quantum Algorithms: Quantum or Classical Problems? 3.3.1 Are there Universal Quantum Algorithms? 3.3.2 Quantum Coin?Deutsch?s Algorithm 3.3.3 Deutsch-Jozsa and Bernstein-Vazirani Algorithms 3.3.4 Shor?s Algorithm 3.3.5 Quantum Simulators 4 Beyond Standard Qubit Computation 4.1 Persistent Entanglement 4.2 One-Way Computing and Cluster States 4.2.1 Dumping Reversibility and Unitarity? 4.2.2 Irreversible Measurement Setup 4.2.3 Reversibility and Unitarity Revisited 4.3 Continuous Variables 4.3.1 Continuous Entanglement 4.3.2 Second-Order (One-Photon) vs. Fourth-Order (Two-Photon) Interference 4.3.3 Qubits with Continuous Spatial Degree of Freedom 4.3.4 Universal Quantum Computation with Continuous-Variable Cluster States 5 Epilogue?Hybrid Systems
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
Today's computers work on bits that exist as either 0 or 1. Quantum computers aren't limited to two states; they encode information as quantum bits, or qubits, which can exist in superposition. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputers. A processor that can use registers of qubits will be able to perform calculations using all the possible values of the input registers simultaneously. This superposition causes a phenomenon called quantum parallelism, and is the motivating force behind the research being carried out in quantum computing. Due to technical obstacles, till date, a quantum computer has not yet been realized. But the concepts and ideas of quantum computing has been demonstrated using various methods like NMR, Ion Trap, Quantum Dot, Optical Methods, etc. A quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single qubit or pair of qubits. In applying these gates in succession, a quantum computer can perform a complicated unitary transformation to a set of qubits in some initial state. The qubits can then be measured, with this measurement serving as the final computational result. Research must devise a way to maintain decoherence and other potential sources of error at an acceptable level. Probably the most important idea in this field is the application of error correction in phase coherence as a means to extract information and reduce error in a quantum system without actually measuring that system. Thereby, quantum computers will emerge as the superior computational devices and perhaps one day make today's modern computer obsolete.
Summary form only given, as follows. Gives a system theoretic approach to the concept of quantum computing. A quantum computer is envisaged to be a system of quantum circuits, acting on a state space, which is a finite-say 2/sup n/-dimensional-complex Hilbert space. The circuitry is a sequence of unitary transforms U/sub t/ /spl epsiv/ SU(2/sup n/) followed by a measurement. These transforms, so-called quantum gates, are controlled by a classical computer, usually especially electromagnetic fields. The state space of a quantum computer has the structure of a Hermitian vector space. Thus it allows "simultaneous" superposition of orthogonal basis states (corresponding to classical states) with the possibility of constructive and destructive interference between different paths of computation. This principle allows the usage of so-called entangled states by preparing the superposition of special "bent" configurations of basis vectors in the product Hilbert space, e.g., as they are known from error control and cryptography. This latter "entanglement" not only promises to make a quantum computer much more powerful than a probabilistic one, but it also allows a method of parallelism. The reason for this lies not only in the fact that the 2/sup n/ dimensional Hilbert space is the n-fold tensor product of 2-dimensional spaces as it is given by spin-1/2-particles, like photons, representing a quantum bit. First of all, we shall address the fact that the power of quantum computing lies in the properties of entangled states as opposed to those of separable states which essentially would resemble classical circuits without giving the feature of exponential speed-up. We shall briefly describe possible physical realisations of quantum (optic, electronic, magnetic,...) devices allowing the engineering of Hamiltonians needed for quantum systems. Basic quantum gates, especially the quantum Horner-Toffoli gate will be discussed to display the principles and methodology for designing quantum circuits. An outlook towards the possible availability of nano- and mesoscopic technologies supporting this new architecture in future generations concludes based on a speculative worst/best case forecast of possible applications, e.g., in public key and conventional cryptography.
Quantum Computation (S. Braunstein). The Los Alamos Trapped Ion Quantum Computer Experiment (R. Hughes, et al.). Experimental Primer on the Trapped Ion Quantum Computer (D. Weineland, et al.). Measurement and State Preparation via Ion Trap Quantum Computing (S. Schneider, et al.). Photon--Wavepackets as Flying Quantum Bits (K. Gheri, et al.). Quantum Logic Gate Operating on Atomic Scattering by Standing Wave Field in Bragg Regime (A. Khan & M. Zubairy). Models of Quantum Turing Machines (P. Benioff). Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing (A. Steane). The Quantum Hamming and Hexacodes (Th. Beth & M. Grassl). Tight Bounds on Quantum Searching (M. Boyer, et al.). Making an Empty Promise with a Quantum Computer (H. Chau & H.--K. Lo). Flocks of Quantum Clones: Multiple Copying of Qubits (V. Buzck, et al.). Information Gain vs. State Disturbance in Quantum Theory (Ch. Fuchs). On Multi--Particle Entanglement (N. Linden & S. Popescu). Generalized Coherent States and Phase--Space--Interference in Multi--Mode Systems (M. Gagen). Index.