t manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere. The Pauli matrices are shown to generate all such unitary transformations. These trans- formations are reversible, because they are characterized by unitary matrices; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable "zoo" of unitary operators, called I, X, Y, Z, H, CNOT, CCNOT, CROSSOVER or SWAP, controlled-U, and controlled- controlled-U. These gates can be used to form quantum circuits, involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product, as progressively introduced through the above description, is eventually formalized. The chapter concludes with the intriguing noncloning theorem, according to which it is not possible to duplicate or "clone" a quantum state. Quantum bits In computer science, the binary digit, or bit, represents the elementary unit of infor- mation. It is a scalar with two possible values, 0 and 1. The bit can also be viewed as a two-state variable, which can be in either the state 0 or the state 1, with no possible other or intermediate states. In Shannon's theory, the bit is also the unit of information, but is a real number satisfying 0    1 which expresses the average information available from a random binary source, which is called entropy. The results of Shan- non's information theory, in fact, concern any information sources, including multi-level or multisymbolic ones, up to any integer number M. In these cases, the bit informa- tion available from the source is bounded according to 0 log2 M. When M is a power of two (M = 2k), it is possible to encode the different symbols into as many 16.1 Quantum bits binary codewords, and the corresponding sources can be conceived as an extension of a binary source. Thus, each bit in a given c