After a short historical overview of the evolution of computers and microelectronics, the author notes that the classical electronic components contained in the actual computers have actually attained the lower limit after which one enters a world dominated by quantum laws.

The first chapter starts with the presentation of the concept of a Turing machine and of Turing computability, and some equivalent definitions of computability. The analysis emphasizes the role of Boolean calculus, and after the introduction of the bit, the elements of Boolean algebra are presented. Next, the author presents the way in which the binary calculus was implemented using transistors and semiconductor circuits, by considering the basic logic gates.

After this preliminary part, the author defines the concept of a quantum bit (qubit) in the context of a two-level system, which is illustrated by a polarized photon with two possible opposite orientations. The concept of transferring quantum information or a transferred qubit is introduced. In the quantum regime, the logical gates are defined, and associations between the numbers and superpositions of qubit states are found.

The Einstein-Podolsky-Rosen paradox and entangled states are presented in a clear manner.

Finally, the author provides an analysis of the teleportation process and presents the basic elements of quantum cryptography.

The second chapter is devoted to the analysis of the possible experimental situations in which quantum computers can be obtained. From the numerous proposals for quantum computing prototypes, a few of the more plausible ones are picked out. The first candidate is based on nuclear magnetic resonance, with molecules in a liquid, and was introduced by Kane. A second valid candidate, proposed by Cirac and Zoller, is based on trapped ion structures. Finally, the requirements necessary for the implementation of quantum computers are discussed.

A similar analysis is made of the requirements necessary for implementation of quantum communication devices. In this field, there is a lack of experiment. However, there are experiments which will be operational after 2012. For these experiments, the future and importance of quantum cryptography and teleportation communications are discussed.

In the third chapter, entitled "Perspectives", first the evolution of the clock speed for classical and quantum computers is discussed. Since quantum computers are based on parallel computers, with parallel CPUs, it is estimated that the speed of calculation will increase exponentially over time, due to an exponential increase in the number of CPUs. In 2012 it is estimated that it will be possible to use $2^{50}$ arrays of bit states.

Single photon sources and the way in which it is possible to maintain a stationary qubit in a well-defined state are discussed.

For three-level systems, having two ground states $g_1$ and $g_2$, close to each other, and an excited level $e$, the author introduces the dark state, i.e., a state $|\Psi^0\rangle=\cos\Theta|g_1\rangle-\sin\Theta|g_2\rangle$, where $\Theta$ is a mixing parameter (angle). The eigenvalue of the dark state $|\Psi^0\rangle$ is zero, and it is not affected by the spontaneous emission from the $e$ state. The author presents the dynamics of the dark states for a cavity and possible teleportation with the aid of dark states. An array of teleportation devices (quantum repeaters) is investigated.

Another problem studied in this chapter is classical to quantum coupling. This study is necessary because the quantum computer must contain macroscopic peripherals, accessible to the human operator.

Finally, quantum algorithms and a short presentation of the basic properties of quantum Turing machines are presented.

The present edition of the book (i.e., ISBN 0-387-24412-3) contains some printing errors. The list of misprints can be obtained from the author. The book contains over 200 references, which contain the most important results from the literature. The author does emphasize in the book his own results.

The book is not a typical treatise on quantum computers and quantum information, but it is an important source for better understanding the basic ideas and results obtained in these fields. The book is accessible and can be useful to physicists, mathematicians and engineers alike.