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The paper is intended to be a survey of all the important aspects and results that have shaped the eld of quantum computation and quantum information. The reader is rst familiarized with those features and principles of quantum mechanics providing a more e cient and secure information processing. Their applications to the general theory of information, cryptography, algorithms, computational complexity and error-correction are then discussed. Prospects for building a practical quantum computer are also analyzed.

The quantum model of computation not only offers entirely new ways to manipulate information, but also allows information processing tasks to be formulated in unconventional, genuine quantum mechanical terms. We show that the task of distinguishing among entangled quantum states combines entanglement and non-determinism in a way that makes the quantum solution impossible to simulate on any classical machine (even one equipped with the same measurement capabilities as the quantum computational device). A new class of information processing tasks is thus uncovered whose members are readily carried out by a quantum computer, yet are impossible to perform on any classical machine (whether deterministic or probabilistic). In the broad, unconventional context created by quantum mechanics, the computational power of a quantum computer is therefore strictly greater than that of a classical computer.

Rank-varying computational complexity describes those computations in which the complexity of executing each step is not a constant, but evolves throughout the computation as a function of the order of execution of each step 2 . This paper identi es practical instances of this computational paradigm in the procedures for computing the quantum Fourier transform and its inverse. It is shown herein that under the constraints imposed by quantum decoherence, only a parallel approach can guarantee a reliable solution or, alternatively, improve scalability.