Quantum Computing

Preface. 1 Introduction and survey. 1.1 Information, computers and quantum mechanics. 1.1.1 Digital information. 1.1.2 Moore's law. 1.1.3 Emergence of quantum behavior. 1.1.4 Energy dissipation in computers. 1.2 Quantum computer basics. 1.2.1 Quantum information. 1.2.2 Quantum communication. 1.2.3 Basics of quantum information processing. 1.2.4 Decoherence. 1.2.5 Implementation. 1.3 History of quantum information processing. 1.3.1 Initial ideas. 1.3.2 Quantum algorithms. 1.3.3 Implementations. 2 Physics of computation. 2.1 Physical laws and information processing. 2.1.1 Hardware representation. 2.1.2 Quantum vs. classical information processing. 2.2 Limitations on computer performance. 2.2.1 Switching energy. 2.2.2 Entropy generation and Maxwell's demon. 2.2.3 Reversible logic. 2.2.4 Reversible gates for universal computers. 2.2.5 Processing speed. 2.2.6 Storage density. 2.3 The ultimate laptop. 2.3.1 Processing speed. 2.3.2 Maximum storage density. 3 Elements of classical computer science. 3.1 Bits of history. 3.2 Boolean algebra and logic gates. 3.2.1 Bits and gates. 3.2.2 2 bit logic gates. 3.2.3 Minimum set of irreversible gates. 3.2.4 Minimum set of reversible gates. 3.2.5 The CNOT gate. 3.2.6 The Toffoli gate. 3.2.7 The Fredkin gate. 3.3 Universal computers. 3.3.1 The Turing machine. 3.3.2 The Church Turing hypothesis. 3.4 Complexity and algorithms. 3.4.1 Complexity classes. 3.4.2 Hard and impossible problems. 4 Quantum mechanics. 4.1 General structure. 4.1.1 Spectral lines and stationary states. 4.1.2 Vectors in Hilbert space. 4.1.3 Operators in Hilbert space. 4.1.4 Dynamics and the Hamiltonian operator. 4.1.5 Measurements. 4.2 Quantum states. 4.2.1 The two dimensional Hilbert space: qubits, spins, and photons. 4.2.2 Hamiltonian and evolution. 4.2.3 Two or more qubits. 4.2.4 Density operator. 4.2.5 Entanglement and mixing. 4.2.6 Quantification of entanglement. 4.2.7 Bloch sphere. 4.2.8 EPR correlations. 4.2.9 Bell's theorem. 4.2.10 Violation of Bell's inequality. 4.2.11 The no cloning theorem. 4.3 Measurement revisited. 4.3.1 Quantum mechanical projection postulate. 4.3.2 The Copenhagen interpretation. 4.3.3 Von Neumann's model. 5 Quantum bits and quantum gates. 5.1 Single qubit gates. 5.1.1 Introduction. 5.1.2 Rotations around coordinate axes. 5.1.3 General rotations. 5.1.4 Composite rotations. 5.2 Two qubit gates. 5.2.1 Controlled gates. 5.2.2 Composite gates. 5.3 Universal sets of gates. 5.3.1 Choice of set. 5.3.2 Unitary operations. 5.3.3 Two qubit operations. 5.3.4 Approximating single qubit gates. 6 Feynman's contribution. 6.1 Simulating physics with computers. 6.1.1 Discrete system representations. 6.1.2 Probabilistic simulations. 6.2 Quantum mechanical computers. 6.2.1 Simple gates. 6.2.2 Adder circuits. 6.2.3 Qubit raising and lowering operators. 6.2.4 Adder Hamiltonian. 7 Errors and decoherence. 7.1 Motivation. 7.1.1 Sources of error. 7.1.2 A counterstrategy. 7.2 Decoherence. 7.2.1 Phenomenology. 7.2.2 Semiclassical description. 7.2.3 Quantum mechanical model. 7.2.4 Entanglement and mixing. 7.3 Error correction. 7.3.1 Basics. 7.3.2 Classical error correction. 7.3.3 Quantum error correction. 7.3.4 Single spin flip error. 7.3.5 Continuous phase errors. 7.3.6 General single qubit errors. 7.3.7 The quantum Zeno effect. 7.3.8 Stabilizer codes. 7.3.9 Fault tolerant computing. 7.4 Avoiding errors. 7.4.1 Basics. 7.4.2 Decoherence free subspaces. 7.4.3 NMR in Liquids. 7.4.4 Scaling considerations. 8 Tasks for quantum computers. 8.1 Quantum versus classical algorithms. 8.1.1 Why Quantum? 8.1.2 Classes of quantum algorithms. 8.2 The Deutsch algorithm: Looking at both sides of a coin at the same time. 8.2.1 Functions and their properties. 8.2.2 Example: one qubit functions. 8.2.3 Evaluation. 8.2.4 Many qubits. 8.2.5 Extensions and generalizations. 8.3 The Shor algorithm: It's prime time. 8.3.1 Some number theory. 8.3.2 Factoring strategy. 8.3.3 The core of Shor's algorithm. 8.3.4 The quantum Fourier transform. 8.3.5 Gates for the QFT. 8.4 The Grover algorithm: Looking for a needle in a haystack. 8.4.1 Oracle functions. 8.4.2 The search algorithm. 8.4.3 Geometrical analysis. 8.4.4 Quantum counting. 8.4.5 Phase estimation. 8.5 Quantum simulations. 8.5.1 Potential and limitations. 8.5.2 Motivation. 8.5.3 Simulated evolution. 8.5.4 Implementations. 9 How to build a quantum computer. 9.1 Components. 9.1.1 The network model. 9.1.2 Some existing and proposed implementations. 9.2 Requirements for quantum information processing hardware. 9.2.1 Qubits. 9.2.2 Initialization. 9.2.3 Decoherence time. 9.2.4 Quantum gates. 9.2.5 Readout. 9.3 Converting quantum to classical information. 9.3.1 Principle and strategies. 9.3.2 Example: Deutsch Jozsa algorithm. 9.3.3 Effect of correlations. 9.3.4 Repeated measurements. 9.4 Alternatives to the network model. 9.4.1 Linear optics and measurements. 9.4.2 Quantum cellular automata. 9.4.3 One way quantum computer. 10 Liquid state NMR quantum computer. 10.1 Basics of NMR. 10.1.1 System and interactions. 10.1.2 Radio frequency field. 10.1.3 Rotating frame. 10.1.4 Equation of motion. 10.1.5 Evolution. 10.1.6 NMR signals. 10.1.7 Refocusing. 10.2 NMR as a molecular quantum computer. 10.2.1 Spins as qubits. 10.2.2 Coupled spin systems. 10.2.3 Pseudo / effective pure states. 10.2.4 Single qubit gates. 10.2.5 Two qubit gates. 10.2.6 Readout. 10.2.7 Readout in multi spin systems. 10.2.8 Quantum state tomography. 10.2.9 DiVincenzo's criteria. 10.3 NMR Implementation of Shor's algorithm. 10.3.1 Qubit implementation. 10.3.2 Initialization. 10.3.3 Computation. 10.3.4 Readout. 10.3.5 Decoherence. 11 Ion trap quantum computers. 11.1 Trapping ions. 11.1.1 Ions, traps and light. 11.1.2 Linear traps. 11.2 Interaction with light. 11.2.1 Optical transitions. 11.2.2 Motional effects. 11.2.3 Basics of laser cooling. 11.3 Quantum information processing with trapped ions. 11.3.1 Qubits. 11.3.2 Single qubit gates. 11.3.3 Two qubit gates. 11.3.4 Readout. 11.4 Experimental implementations. 11.4.1 Systems. 11.4.2 Some results. 11.4.3 Problems. 12 Solid state quantum computers. 12.1 Solid state NMR/EPR. 12.1.1 Scaling behavior of NMR quantum information processors. 12.1.2 31P in silicon. 12.1.3 Other proposals. 12.1.4 Single spin readout. 12.2 Superconducting systems. 12.2.1 Charge qubits. 12.2.2 Flux qubits. 12.2.3 Gate operations. 12.2.4 Readout. 12.3 Semiconductor qubits. 12.3.1 Materials. 12.3.2 Excitons in quantum dots. 12.3.3 Electron spin qubits. 13 Quantum communication. 13.1 Quantum only tasks. 13.1.1 Quantum teleportation. 13.1.2 (Super ) Dense coding. 13.1.3 Quantum key distribution. 13.2 Information theory. 13.2.1 A few bits of classical information theory. 13.2.2 A few bits of quantum information theory. Appendix. A. Two spins 1/2: Singlet and triplet states. B. Symbols and abbreviations. Bibliography. Index.