Progress in analytical investigations of the achievement of fault tolerance in quantum computing

We describe progress made in understanding and assuring fault tolerance in quantum computation. We introduce and explore analytical techniques for explicitly determining the logical state of a quantum computer undergoing dynamical evolution according to an arbitrary quantum algorithm. We carry out detailed analyses of the effects of errors, paying special attention to the general case of non-equiprobable errors, i.e., the important and realistic situation in which the probabilities for sigma\textunderscore x, sigma\textunderscore y and sigma\textunderscore z errors are not necessarily the same (sigma\textunderscore x, sigma\textunderscore y and sigma\textunderscore z are the Pauli operators).