Introduction to Quantum Computers

Quantum computing promises to solve problems which are intractable on digital computers. Highly parallel quantum algorithms can decrease the computational time for some problems by many orders of magnitude. This important book explains how quantum computers can do these amazing things. Several algorithms are illustrated: the discrete Fourier transform, Shor's algorithm for prime factorization; algorithms for quantum logic gates; physical implementations of quantum logic gates in ion traps and in spin chains; the simplest schemes for quantum error correction; correction of errors caused by imperfect resonant pulses; correction of errors caused by the nonresonant actions of a pulse; and numerical simulations of dynamical behavior of the quantum Control-Not gate. An overview of some basic elements of computer science is presented, including the Turing machine, Boolean algebra, and logic gates. The required quantum ideas are explained. Contents 8 Chapter 1 Introduction 10 Chapter 2 The Turing Machine 17 Chapter 3 Binary System and Boolean Algebra 22 Chapter 4 The Quantum Computer 29 Chapter 5 The Discrete Fourier Transform 40 Chapter 6 Quantum Factorization of Integers 45 Chapter 7 Logic Gates 47 Chapter 8 Implementation of Logic Gates Using Transistors 53 Chapter 9 Reversible Logic Gates 60 Chapter 10 Quantum Logic Gates 68 Chapter 11 Two and Three Qubit Quantum Logic Gates 73 Chapter 12 One-Qubit Rotation 78 Chapter 13 Aj – Transformation 87 Chapter 14 B jk – Transformation 92 Chapter 15 Unitary Transformations and Quantum Dynamics 94 Chapter 16 Quantum Dynamics at Finite Temperature 99 Chapter 17 Physical Realization of Quantum Computations 110 Chapter 18 CONTROL-NOT Gate in an Ion Trap 118 Chapter 19 A j and B jk Gates in an Ion Trap 125 Chapter 20 Linear Chains of Nuclear Spins 129 Chapter 21 Digital Gates in a Spin Chain 133 Chapter 22 Non-resonant Action of - Pulses 136 Chapter 23 Experimental Logic Gates in Quantum Systems 145 Chapter 24 Error Correction for Quantum Computers 152 Chapter 25 Quantum Gates in a Two-Spin System 163 Chapter 26 Quantum Logic Gates in a Spin Ensemble at Room Temperature 169 Chapter 27 Evolution of an Ensemble of Four-Spin Molecules 176 Chapter 28 Getting the Desired Density Matrix 183 Chapter 29 Conclusion 187 Bibliography 189 The Turing Machine. The Quantum Computer. Logic Gates. 1. Introduction -- 2. The Turing Machine -- 3. Binary System And Boolean Algebra -- 4. The Quantum Computer -- 5. The Discrete Fourier Transform -- 6. Quantum Factorization Of Integers -- 7. Logic Gates -- 8. Implementation Of Logic Gates Using Transistors -- 9. Reversible Logic Gates -- 10. Quantum Logic Gates -- 11. Two And Three Qubit Quantum Logic Gates -- 12. One-qubit Rotation -- 13. A[subscript J] -- Transformation -- 14. B[subscript Jk] -- Transformation -- 15. Unitary Transformations And Quantum Dynamics -- 16. Quantum Dynamics At Finite Temperature -- 17. Physical Realization Of Quantum Computations -- 18. Control-not Gate In An Ion Trap -- 19. A[subscript J] And B[subscript Jk] Gates In An Ion Trap. Gennady P. Berman ... [et Al.]. Includes Bibliographical References (p. 180-184) And Index. Quantum computing promises to solve problems which are intractable on digital computers. Highly parallel quantum algorithms can decrease the computational time for some problems by many orders of magnitude. This important book explains how quantum computers can do these amazing things. Several algorithms are illustrated: the discrete Fourier transform, Shorâ€TMs algorithm for prime factorization; algorithms for quantum logic gates; physical implementations of quantum logic gates in ion traps and in spin chains; the simplest schemes for quantum error correction; correction of errors caused by im Quantum computing promises to solve problems which are intractable on digital computers. This book explains how quantum computators can do these amazing things. Several algorithms are illustrated, including the discrete Fourier transform and Shor's algorithm for prime factorization.