## Classical and Quantum ComputationThis book is an introduction to a new rapidly developing theory of quantum computing. It begins with the basics of classical theory of computation: Turing machines, Boolean circuits, parallel algorithms, probabilistic computation, NP-complete problems, and the idea of complexity of an algorithm. The second part of the book provides an exposition of quantum computation theory. It starts with the introduction of general quantum formalism (pure states, density matrices, and superoperators), universal gate sets and approximation theorems. Then the authors study various quantum computation algorithms: Grover's algorithm, Shor's factoring algorithm, and the Abelian hidden subgroup problem. In concluding sections, several related topics are discussed (parallel quantum computation, a quantum analog of NP-completeness, and quantum error-correcting codes).Rapid development of quantum computing started in 1994 with a stunning suggestion by Peter Shor to use quantum computation for factoring large numbers - an extremely difficult and time-consuming problem when using a conventional computer. Shor's result spawned a burst of activity in designing new algorithms and in attempting to actually build quantum computers. Currently, the progress is much more significant in the former: a sound theoretical basis of quantum computing is under development and many algorithms have been suggested.In this concise text, the authors provide solid foundations to the theory - in particular, a careful analysis of the quantum circuit model - and cover selected topics in depth. Included are a complete proof of the Solovay-Kitaev theorem with accurate algorithm complexity bounds, approximation of unitary operators by circuits of doubly logarithmic depth. Among other interesting topics are toric codes and their relation to the anyon approach to quantum computing. Prerequisites are very modest and include linear algebra, elements of group theory and probability, and the notion of a formal or an intuitive algorithm. This text is suitable for a course in quantum computation for graduate students in mathematics, physics, or computer science. More than 100 problems (most of them with complete solutions) and an appendix summarizing the necessary results are a very useful addition to the book. It is available in both hardcover and softcover editions. |

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### Contents

1 | |

Classical Computation | 9 |

Quantum Computation | 53 |

Solutions | 177 |

Appendix A Elementary Number Theory | 237 |

Bibliography | 251 |

### Other editions - View all

Classical and Quantum Computation Alexei Yu. Kitaev,Alexander Shen,Mikhail N. Vyalyi No preview available - 2002 |

Classical and Quantum Computation Alexei Yu. Kitaev,Alexander Shen,Mikhail N. Vyalyi No preview available - 2002 |

Classical and Quantum Computation Alexei Yu. Kitaev,Alexander Shen,Mikhail N. Vyalyi No preview available - 2002 |

### Common terms and phrases

ancillas apply approximate arbitrary assume basis vectors binary bits Boolean circuit Boolean function classical complete basis complexity conﬁguration consider controlling qubit copies corresponding deﬁned deﬁnition denotes density matrix depth eigenvalues elements encoding equation error example exists factor ﬁeld ﬁnd ﬁnding ﬁnite ﬁrst ﬁxed follows formula fraction function F gate graph Hamiltonian inequality input string integer Lemma length linear measuring operator mod q multiplication nonnegative nonzero norm Note NP-complete obtain operator norm oracle output P/poly pair partial trace path permutation physically realizable poly(n polynomial precision predicate prime probabilistic problem proof Prove PSPACE quantum algorithm quantum circuit quantum computation qubits random represented result reversible circuit satisﬁes sequence simulation solution space Speciﬁcally standard basis steps SU(M subgroup subspace sufﬁcient superoperator symbol symplectic tape Theorem Toffoli gate transformation Turing machine unitary operator vertex vertices Z/qZ