## Principles of Quantum Computation and Information, Volume 1Quantum computation and information is a new, rapidly developing interdisciplinary field. Therefore, it is not easy to understand its fundamental concepts and central results without facing numerous technical details. This book provides the reader a useful and not-too-heavy guide. It offers a simple and self-contained introduction; no previous knowledge of quantum mechanics or classical computation is required.Volume I may be used as a textbook for a one-semester introductory course in quantum information and computation, both for upper-level undergraduate students and for graduate students. It contains a large number of solved exercises, which are an essential complement to the text, as they will help the student to become familiar with the subject. The book may also be useful as general education for readers who want to know the fundamental principles of quantum information and computation and who have the basic background acquired from their undergraduate course in physics, mathematics, or computer science. |

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### Contents

Introduction | 1 |

Introduction to Classical Computation | 9 |

111 Addition on a Turing machine | 12 |

112 The ChurchTuring thesis | 13 |

113 The universal Turing machine | 14 |

115 The halting problem | 15 |

121 Binary arithmetics | 17 |

123 Universal classical computation | 22 |

37 Function evaluation | 132 |

38 The quantum adder | 137 |

39 Deutschs algorithm | 140 |

391 The DeutschJozsa problem | 141 |

392 An extension of Deutschs algorithm | 143 |

310 Quantum search | 144 |

3101 Searching one item out of four | 145 |

3102 Searching one item out of N | 148 |

13 Computational complexity | 24 |

131 Complexity classes | 27 |

132 The Chernoff bound | 30 |

141 Deterministic chaos | 31 |

142 Algorithmic complexity | 33 |

15 Energy and information | 35 |

152 Londoners principle | 37 |

153 Extracting work from information | 40 |

16 Reversible computation | 41 |

161 Toffoli and Fredkin gates | 43 |

162 The billiardball computer | 45 |

17 A guide to the bibliography | 47 |

Introduction to Quantum Mechanics | 49 |

21 The SternGerlach experiment | 50 |

22 Youngs doubleslit experiment | 53 |

23 Linear vector spaces | 57 |

24 The postulates of quantum mechanics | 76 |

25 The EPR paradox and Bells inequalities | 88 |

26 A guide to the bibliography | 97 |

Quantum Computation | 99 |

31 The qubit | 100 |

311 The Bloch sphere | 102 |

312 Measuring the state of a qubit | 103 |

32 The circuit model of quantum computation | 105 |

33 Singlequbit gates | 108 |

331 Rotations of the Bloch sphere | 110 |

34 Controlled gates and entanglement generation | 112 |

341 The Bell basis | 118 |

351 Preparation of the initial state | 127 |

36 Unitary errors | 130 |

3103 Geometric visualization | 149 |

311 The quantum Fourier transform | 152 |

312 Quantum phase estimation | 155 |

313 Finding eigenvalues and eigenvectors | 158 |

314 Period finding and Shors algorithm | 161 |

315 Quantum computation of dynamical systems | 164 |

3152 The quantum bakers map | 168 |

3153 The quantum sawtooth map | 170 |

3154 Quantum computation of dynamical localization | 174 |

316 First experimental implementations | 178 |

3161 Elementary gates with spin qubits | 179 |

3162 Overview of the first implementations | 181 |

317 A guide to the bibliography | 185 |

Quantum Communication | 189 |

411 The Vernam cypher | 190 |

412 The publickey crypto system | 191 |

413 The RSA protocol | 192 |

42 The nocloning theorem | 194 |

421 Fasterthanlight transmission of information? | 197 |

43 Quantum cryptography | 198 |

431 The BB84 protocol | 199 |

432 The E91 protocol | 202 |

44 Dense coding | 205 |

45 Quantum teleportation | 208 |

46 An overview of the experimental implementations | 213 |

47 A guide to the bibliography | 214 |

Solutions to the exercises | 215 |

241 | |

253 | |

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### Common terms and phrases

Alice and Bob ancillary qubits angle atoms axis baker's map Bell's inequalities Bloch sphere circuit of Fig classical bits classical computer CNOT gate complex numbers computational basis control qubit corresponding defined denotes discussed easy to check efficiently eigenstate eigenvalues eigenvectors elementary gates energy entangled EPR pair error Exercise experimental exponentially given Grover's algorithm Hadamard gates Hamiltonian Hermitian Hilbert space input integer large number Let us consider matrix representation measurement molecule n-bit n-qubit Note number of qubits obtain outcome output particle Pauli performed phase-shift gates photon physical polarization polynomial possible principle probabilistic probability problem protocol quantum algorithm quantum circuit quantum circuit implementing quantum computer quantum Fourier transform quantum gates quantum mechanics quantum systems qubit rotation sawtooth map Schrodinger equation secret key sequence simulation single single-qubit solution solve spin Stern-Gerlach superposition target qubit teleportation Toffoli gate Turing machine two-qubit unitary operator unitary transformation wave function