  The main ideas of quantum error correction are introduced. These are
encoding, extraction of syndromes, error operators, and code construction. It
is shown that general noise and relaxation of a set of 2-state quantum systems
can always be understood as a combination of Pauli operators acting on the
system. Each quantum error correcting code allows a subset of these errors to
be corrected. In many situations the noise is such that the remaining
uncorrectable errors are unlikely to arise, and hence quantum error correction
has a high probability of success. In order to achieve the best noise tolerance
in the presence of noise and imprecision throughout the computer, a
hierarchical construction of a quantum computer is proposed.
