GEOMETRY OF QUANTUM COMPUTATION ; GEOMETRY OF QUANTUM COMPUTATION ; Contents; Preface; Basics of Quantum Computation; 1.1. The Space of Quantum Bits; 1.2. Logical Gates for Quantum Computer; 1.3 .Construction of Single Qubit Gates; 1.4. The Entangled Operator; 1.5. Geometric Character for Quantum Computing; 1.5.1. Adiabatic Theorem; 1.5.2. Holonomy and Monodromy Operators; Quantum Computers Based on Exactly Solvable Models and Geometric Phases; 2.1. Construction of a Time-Dependent Hamiltonian; 2.2. Geometric Phases; 2.3. Construction of an Universal Set of Gates
Quantum Processor Based on the Three-Level Quantum System3.1. Atom-photon Interactions Hamiltonian; 3.2. Simpli cation of Atom-Photon Interactions Hamiltonian; 3.2.1. Selection the Real Transitions and Rotating Wave Approximation; 3.2.2. Interaction Representation; 3.3. The Quantum Dynamics of a Three-Level Atom; 3.4. Calculation of the Time Evolution Operator and Unitary Transformations; 3.5. Probability Amplitude Method; 3.6. Control of Three-Leve lQuantum System; Methods of Geometric Control Theory for Quantum Computations; 4.1.Basic Notation from Geometric Control Theory
4.1.1. Dynamical Polysystems and Control Group4.1.2. The Structure of an Orbit o fDynamical Polysystem; 4.1.3. Control System and Accessibility Problem; 4.1.4. Two Notions of Controllability; 4.2. Control Systems on Lie Groups; 4.2.1. Homogeneous and Af ne Cases; 4.2.2. Control system on Compact Lie Group with Continuous Hamiltonian; 4.2.3. Minimal Set of Generators for su(n); 4.3. Recursive Construction o fQuantum Gates; 4.4. Vector Bundle with Connection on Grassmann Manifold; 4.4.1. The Differential Structure on Complex Grassmann Manifold
5.1. Fuchsian System and Vector Bundle on the Riemann Sphere5.2. Schr odinger Equations and Fuchsian Systems; 5.3. Monodromy Approach to Quantum Computing; 5.4. Scattering Matrices as the Gates for Quantum Computer; 5.4.1. Scattering on the Line and Universal Gates; 5.4.2. Two-Level System in Electric Field; 5.4.3. The Fuchsian Systems Monodromy and S-Matrix; 5.5. Quantum Monodromy; References; Index
Bibliography
Includes bibliographical references (pages 165-174) and index
