## Quantum Computers, Algorithms, and ChaosGiulio Casati, Dima L. Shepelyansky, Peter Zoller, Giuliano Benenti "During the last ten years Quantum Information Processing and Communication (QIPC) has established itself as one of the new hot topic fields in physics, with the potential to revolutionize many areas of science and technology. QIPC replaces the laws of classical physics applied to computation and communication with the more fundamental laws of quantum mechanics. This becomes increasingly important due to technological progress going down to smaller and smaller scales where quantum effects start to be dominant. In addition to its fundamental nature, QIPC promises to advance computing power beyond the capabilities of any classical computer, to guarantee secure communication and establish direct links to emerging quantum technologies, such as, for example, quantum based sensors and clocks. One of the outstanding feature of QIPC is its interdisciplinary character: it brings together researchers from physics, mathematics and computer science. In particular, within physics we have seen the emergence of a new QIPC community, which ranges from theoretical to experimental physics, and crosses boundaries of traditionally separated disciplines such as atomic physics, quantum optics, statistical mechanics and solid state physics, all working on different and complementary aspects of QIPC. This publication covers the following topics: Introduction to quantum computing; Quantum logic, information and entanglement; Quantum algorithms; Error-correcting codes for quantum computations; Quantum measurements and control; Quantum communication; Quantum optics and cold atoms for quantum information; Quantum computing with solid state devices; Theory and experiments for superconducting qubits; Interactions in many-body systems: quantum chaos, disorder and random matrices; Decoherence effects for quantum computing; and Flature prospects of quantum information processing." |

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### Contents

STEANE A tutorial on quantum error correction | 1 |

GRASSL Encoding and decoding quantum errorcorrecting codes | 33 |

Linear optics quantum computation | 60 |

LOQC and quantum error correction | 76 |

TOMBESI Entanglement in quantum optics | 95 |

Continuous variable systems | 101 |

Continuous variable entanglement | 109 |

HEIN W DUR J EISERT R RAUSSENDORF M VAN DEN NEST | 116 |

Loss of entanglement by dephasing | 329 |

Threequbit entanglement | 336 |

Bell inequality with noise correlators | 342 |

G VlDAL Entanglement and matrix product states in onedimensional | 349 |

H MABUCHI Applications of quantum filtering and feedback | 383 |

G FALCI and R FAZIO Quantum computation with Josephson qubits | 393 |

nanocircuits | 430 |

G ITHIER F NGUYEN E COLLIN N BOULANT P J MEESON | 447 |

Definitions for graph states | 124 |

Clifford operations and classical simulation | 148 |

Physical implementations | 160 |

Entanglement in graph states | 175 |

B GEORGEOT Quantum algorithms and quantum chaos | 219 |

VII | 265 |

G BENENTI and G CASATI Quantum chaos decoherence and quantum | 267 |

Effects of imperfections in the quantum computer hardware | 280 |

Quantum noise and quantum trajectories | 291 |

Final remarks | 304 |

Entanglement basics | 310 |

Spin vs orbital entanglement | 322 |

The quantronium circuit | 456 |

ELZERMAN L P KOUWENHOVEN and L M K VANDERSYPEN | 471 |

EsCHNER Quantum computation with trapped ions | 499 |

Experimental techniques | 509 |

BLOCK Engineering multiparticle entanglement with neutral atoms | 521 |

BoseHubbard model of interacting bosons in optical lattices | 529 |

Collapse and revival of a macroscopic quantum field | 536 |

Entanglement generation via spin changing collisions | 545 |

N DAVIDSON A KAPLAN M F ANDERSEN and T GRUNZWEIG Clas | 555 |

Elenco dei partecipanti | 596 |

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amplitude applied atoms basis beam splitter bi-partite billiard binary chaotic circuit classical computation coherent consider Cooper pair correlation corresponding coupling decay decoherence defined density matrix dephasing described detector dynamics efficiently eigenstates electron encoded energy entanglement equivalent evolution example exponential fidelity Fourier transform frequency Gaussian graph G Hamiltonian Hilbert space implemented initial interaction iterations Josephson laser lattice Lett linear optics maximal mode momentum noise number of qubits obtained operators oscillations pair parameters particles Pauli perturbation phase space photons Phys Physics polynomial potential probability protocol pulse quantum algorithm quantum chaos quantum computer quantum dot quantum error correction quantum Fourier transform quantum gates quantum information quantum register quantum systems regime Schmidt measure Schmidt rank shown in fig simulation spin stabilizer subsect subspace teleportation trajectories trap tunnel two-qubit unitary vector vertex vertices voltage wave function weighted graph Wigner function