Quantum computing promises to solve problems which are intractable on digital computers. Highly parallel quantum algorithms can decrease the computational time for some problems by many orders of magnitude. This important book explains how quantum computers can do these amazing things. Several algorithms are illustrated: the discrete Fourier transform, Shor's algorithm for prime factorization; algorithms for quantum logic gates; physical implementations of quantum logic gates in ion traps and in spin chains; the simplest schemes for quantum error correction; correction of errors caused by imperfect resonant pulses; correction of errors caused by the nonresonant actions of a pulse; and numerical simulations of dynamical behavior of the quantum Control-Not gate. An overview of some basic elements of computer science is presented, including the Turing machine, Boolean algebra, and logic gates. The required quantum ideas are explained.
The Turing machine; binary system and Boolean algebra; the quantum computer; the discrete Fourier transform; quantum factorization of integers; logic gates; implementation of logic gates using transistors; reversible logic gates; quantum logic gates; tow and three Qubit quantum logic gates; on-Qubit rotation; Aj transformation; Bjk transformation; unitary transformations and quantum dynamics; quantum dynamics at finite temperature; physical realization of quantum computations; Control-Not gate in an ion trap; Aj and Bjk gates in an ion trap; linear chains of nuclear spins; digital gates in a spin chain; nonresonant action of pi-pulses; experimental logic gates in quantum systems - achievements and opportunities; error correction for quantum gates in a two-spin system; quantum logic gates in a spin ensemble at room temperature; evolution of an ensemble of four-spin molecules; how to get the desired density matrix?