*WILEY- Quantum Computing. Joachim Stolze and Dieter Suter. A Short Course from Theory to Experiment. WILEY-VCH Verlag GmbH & Co.

Transcription

1 Joachim Stolze and Dieter Suter Quantum Computing A Short Course from Theory to Experiment Second, Updated and Enlarged Edition *WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

2 Contents Preface XIII 1 Introduction and Survey Information, Computers, and Quantum Mechanics Digital Information Moore's Law Emergence of Quantum Behavior Energy Dissipation in Computers Quantum Computer Basics Quantum Information Quantum Communication Basics of Quantum Information Processing Dec oherence Implementation History of Quantum Information Processing Initial Ideas Quantum Algorithms Implementations Physics of Computation Physical Laws and Information Processing Hardware Representation Quantum vs. Classical Information Processing Limitations on Computer Performance Switching Energy Entropy Generation and Maxwell's Demon Reversible Logic Reversible Gates for Universal Computers Processing Speed Storage Density The Ultimate Laptop Processing Speed Maximum Storage Density... 24

3 VI Contents 3 Elements of Classical Computer Science Bits of History Boolean Algebra and Logic Gates Bits and Gates Bit Logic Gates Minimum Set of Irreversible Gates Minimum Set of Reversible Gates The CNOT Gate The Toffoli Gate The Fredkin Gate Universal Computers The Turing Machine The Church Turing Hypothesis Complexity and Algorithms Complexity Classes Hard and Impossible Problems Quantum Mechanics General Structure Spectral Lines and Stationary States Vectors in Hüben Space Operators in Hilbert Space Dynamics and the Hamiltonian Operator Measurements Quantum States The Two-Dimensional Hilbert Space: Qubits, Spins, and Photons Hamiltonian and Evolution Coupling to Environment Density Operator Entanglement and Mixing Quantification of Entanglement Bloch Sphere EPR Correlations Bell's Theorem Violation of Bell's Inequality The No-Cloning Theorem Measurement Revisited Quantum Mechanical Projection Postulate The Copenhagen Interpretation Von Neumann's Model Quantum Bits and Quantum Gates Single-Qubit Gates Introduction Rotations Around Coordinate Axes... 65

4 Contents VII General Rotations Composite Rotations... Two-Qubit Gates Controlled Gates Composite Gates... Universal Sets of Gates Choice of Set Unitary Operations Two-Qubit Operations Approximating Single-Qubit Gates Feynman's Contribution Simulating Physics with Computers Discrete System Representations Probabilistic Simulations Quantum Mechanical Computers Simple Gates Adder Circuits Qubit Raising and Lowering Operators Adder Hamiltonian Errors and Decoherence Motivation Sources of Error A Counterstrategy Decoherence Phenomenology Semiclassical Description Quantum Mechanical Model Entanglement and Mixing Error Correction Basics Classical Error Correction Quantum Error Correction Single Spin-Flip Error Continuous Phase Errors General Single Qubit Errors The Quantum Zeno Effect Stabilizer Codes Fault-Tolerant Computing Avoiding Errors Basics Decoherence-Free Subspaces NMR in Liquids Scaling Considerations

5 VIII Contents 8 Tasks for Quantum Computers Quantum Versus Classical Algorithms Why Quantum? Classes of Quantum Algorithms The Deutsch Algorithm: Looking at Both Sides of a Coin at the Same Time Functions and Their Properties Example: One-Qubit Functions Evaluation Many Qubits Extensions and Generalizations The Shor Algorithm: It's Prime Time Some Number Theory Factoring Strategy The Core of Shor's Algorithm The Quantum Fourier Transform Gates for the QFT The Grover Algorithm: Looking for a Needle in a Haystack Oracle Functions The Search Algorithm Geometrical Analysis Quantum Counting Phase Estimation Quantum Simulations Potential and Limitations Simulated Evolution Implementations How to Build a Quantum Computer Components The Network Model Some Existing and Proposed Implementations Requirements for Quantum Information Processing Hardware Qubits Initialization Decoherence Time Quantum Gates Readout Converting Quantum to Classical Information Principle and Strategies Example: Deutsch Jozsa Algorithm Effect of Correlations Repeated Measurements Alternatives to the Network Model '14.1 Linear Outics and Measurements

6 Contents IX Quantum Cellular Automata One-Way Quantum Computer Liquid State NMR Quantum Computer Basics of NMR System and Interactions Radio Frequency Field Rotating Frame Equation of Motion Evolution NMR Signals Refocusing NMR as a Molecular Quantum Computer Spins as Qubits Coupled Spin Systems Pseudo/Effective Pure States Single-Qubit Gates Two-Qubit Gates Readout Readout in Multispin Systems Quantum State Tomography DiVincenzo's Criteria NMR Implementation of Shor's Algorithm Qubit Implementation Initialization Computation Readout Decoherence Trapped Ions and Atoms Trapping Ions Ions, Traps, and Light Linear Traps Interaction with Light Optical Transitions Motional Effects Basics of Laser Cooling Quantum Information Processing with Trapped Ions Qubits Single-Qubit Gates Two-Qubit Gates Readout Experimental Implementations I Systems Some Results Challenges

7 X Contents 11.5 Neutral Atoms Trapping Neutral Particles Manipulating Neutral Particles Gate Operations Interacting Atoms in Optical Lattices Interacting Particles in a Periodic Potential: The Hubbard Model (Observing) The Mott Hubbard Transition Universal Optical Lattice Quantum Computing? Solid-State Quantum Computers Solid State NMR/EPR Scaling Behavior of NMR Quantum Information Processors in Silicon Other Proposals Single-Spin Readout Superconducting Systems Charge Qubits Flux Qubits Gate Operations Readout Semiconductor Qubits Materials Excitons in Quantum Dots Electron Spin Qubits Photons for Quantum Information "Quantum Only" Tasks Quantum Teleportation (Super-) Dense Coding Quantum Key Distribution A Few Bits of Classical Information Theory Measuring Information Information Content and Entropy Mutual Information and the Data Processing Inequality Data Compression and Shannon's Noiseless Channel Coding Theorem The Binary Symmetric Channel and Shannon's Noisy Channel Coding Theorem A Few Bits of Quantum Information Theory The von Neumann Entropy The Accessible Information and Holevo's Bound Schumacher's Noiseless Channel Coding Theorem Classical Information over Noisy Quantum Channels

8 Contents XI Appendix A Two Spins-1/2: Singlet and Triplet States 237 Bibliography 239 Index 261