Quantum Computing is a relatively young field found at the intersection between information science and quantum physics. It describes the manipulation of information stored in the quantum states of a physical system, which has properties that sharply contrast with those of conventional 'classical' information. The potential for exploitation of these properties - quantum information processing - is huge. Quantum algorithms running on the holy grail of QI research, a quantum computer, could be faster than any forseeable classical computer.Quantum Computing - from Linear Algebra to Physical Realizations is a complete introduction to the subject, based on courses given by the authors in both Japan and Europe. The text divides into two parts - the first, an introductory overview of quantum information theory, including a discussion of the famous Grover and Shor algorithms and the introduction of the concept of a qubit (quantum-bit). The second part - which makes this title unique - is a comprehensive discussion of the experimental approaches currently being developed to make quantum computing a reality. These range from optical techniques (already successfully used for quantum cryptography) through solid state / superconducting and BEC methods to ion traps - the most likely candidate for eventual success.