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1 Bits and Registers 1 1.1 Physical Embodiments of a Bit 1 1.2 Registers 4 1.3 Fluctuating Registers 8 1.4 Mixtures and Pure States 13 1.5 Basis States 19 1.6 Functions and Measurements on Mixtures 21 1.7 Forms and Vectors 28 1.8 Transformations of Mixtures 32 1.9 Composite Systems 37 2 The Qubit 41 2.1 The Evil Quanta 41 2.2 The Fiducial Vector of a Qubit 47 2.3 The Stern-Gerlach Experiment 49 2.4 Polarized States 55 2.5 Mixtures of Qubit States 59 2.6 The Measurement 62 2.7 Pauli Vectors and Pauli Forms 65 2.8 The Hamiltonian Form 69 2.9 Qubit Evolution 72 2.10 Larmor Precession 75 2.11 Rabi Oscillations 78 2.11.1 Solution at Resonance 80 2.11.2 Solution off Resonance 85 2.12 The Quantronium 86 3 Quaternions 95 3.1 Continuing Development of Quantum Mechanics 95 3.2 Hamilton Quaternions 96 3.3 Pauli Quaternions 97 3.4 From Fiducial Vectors to Quaternions 99 3.5 Expectation Values 100 3.6 Mixtures 103 3.7 Qubit Evolution 104 3.8 Why Does It Work? 107 4 The Unitary Formalism 111 4.1 Unpacking Pauli Quaternions 111 4.2 Pauli Matrices 111 4.3 The Basis Vectors and the Hilbert Space 117 4.4 The Superstition of Superposition 121 4.5 Probability Amplitudes 130 4.6 Spinors 134 4.7 Operators and Operands 142 4.8 Properties of the Density Operator 148 4.9 Schr6dinger Equation 151 4.9.1 General Solution of the Schr6dinger Equation 153 4.9.2 Larmor Precession Revisited 160 4.10 Single Qubit Gates 162 4.11 Taking Qubits for a Ride 168 4.11.1 Dragging a Qubit along an Arbitrary Trajectory 169 4.11.2 Closed Trajectory Case 174 4.11.3 A Qubit in the Rotating Magnetic Field 177 4.11.4 Observing the Berry Phase Experimentally 183 4.11.5 Berry Phase Gates 187 5 The Biqubit 191 5.1 Entangled States 191 5.2 Pauli Exclusion Principle 204 5.3 A Superconducting Biqubit 206 5.4 An Atom and a Photon 216 5.5 A Biqubit in a Rotated Frame 219 5.6 Bell Inequality 224 5.7 Nonlocality 230 5.8 Single-Qubit Expectation Values 233 5.9 Classification of Biqubit States 236 5.10 Separability 247 5.11 Impure Quantum Mechanics 259 5.11.1 Nonunitary Evolution 262 5.11.2 Depolarization 266 5.11.3 Dephasing 270 5.11.4 Spontaneous Emission 273 5.12 Schr6dinger's Cat 277 5.12.1 The Haroche-Ramsey Experiment 279 6 The Controlled NOT Gate 289 6.1 The Quintessence of Quantum Computing 289 6.2 Universal Gates 292 6.2.1 The ABC Decomposition and Controlled- U 292 6.2.2 General Biqubit Unitary Gates 296 6.2.3 Triqubit Gates 303 6.2.4 Universality of the Deutsch Gate 306 6.3 The Cirac-Zoller Gate 322 6.4 The Superconducting Gate 333 7 Yes, It Can Be Done with Cogwheels 343 7.1 The Deutsch Oracle 343 7.2 NMR Computing 352 7.3 Brassard Teleportation Circuit 362 7.4 The Grover Search Algorithm 371 7.5 Cogwheels 376 7.6 The Crossroad 387 A Quaternions and Pauli Matrices 393 A.1 Hamilton Quaternions 393 A.2 Pauli Quaternions 393 A.3 Pauli Matrices 395 B Biqubit Probability Matrices 397 C Tensor Products of Pauli Matrices 399