One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.

2.4 The density operator 98 2.4.1 Ensembles of quantum states 99 2.4.2 General properties of the density operator 101 2.4.3 The reduced density operator 105 2.5 The Schmidt decomposition and purifications 109 2.6 EPR and the Bell inequality 111

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Decoherence in quantum computers is formulated within the Semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for error-less quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and moreover, that universal quantum computation is possible within them.

Quantum computation remains an enormously appealing but elusive goal. It is appealing because of its potential to perform superfast algorithms, such as finding prime factors in polynomial time, but also elusive because of the difficulty of simultaneously manipulating quantum degrees of freedom while preventing environmentally induced decoherence. A new approach to quantum computing is introduced based on the use of multiple-pulse resonance techniques to manipulate the small deviation from equilibrium of the density matrix of a macroscopic ensemble so that it appears to be the density matrix of a much lower dimensional pure state. A complete prescription for quantum computing is given for such a system.