  In this tutorial we review physical implementation of quantum computing using
a system of cold trapped ions. We discuss systematically all the aspects for
making the implementation possible. Firstly, we go through the loading and
confining of atomic ions in the linear Paul trap, then we describe the
collective vibrational motion of trapped ions. Further, we discuss interactions
of the ions with a laser beam. We treat the interactions in the travelling-wave
and standing-wave configuration for dipole and quadrupole transitions. We
review different types of laser cooling techniques associated with trapped
ions. We address Doppler cooling, sideband cooling in and beyond the Lamb-Dicke
limit, sympathetic cooling and laser cooling using electromagnetically induced
transparency. After that we discuss the problem of state detection using the
electron shelving method. Then quantum gates are described. We introduce
single-qubit rotations, two-qubit controlled-NOT and multi-qubit controlled-NOT
gates. We also comment on more advanced multi-qubit logic gates. We describe
how quantum logic networks may be used for the synthesis of arbitrary pure
quantum states. Finally, we discuss the speed of quantum gates and we also give
some numerical estimations for them. A discussion of dynamics on off-resonant
transitions associated with a qualitative estimation of the weak coupling
regime and of the Lamb-Dicke regime is included in Appendix.
