With explicit derivations, exercises, and selected solutions, this self-contained, classroom-tested book examines the theoretical aspects of quantum computing and focuses on several candidates of a working quantum computer, evaluating them according to the DiVincenzo criteria. It covers theoretical tools, such as vectors, matrices, quantum gates, and integral and Fourier transforms. The authors discuss how decoherence is an obstacle to the physical realization of a working quantum computer and how quantum error-correcting codes can sometimes overcome this. They also explore applications, including the ion trap quantum computer and the solid state realization of a quantum computer, at the forefront of the field.