While they capture essential features of noise in a quantum-mechanical setting, the assumptions that go into them-such as locality and stochastic independence of error events-are not obeyed exactly in any physical system of interest. It shall be pointed out from the beginning that the decoherence models discussed in this study are phenomenological and thus overly simplistic. Woven together, and supplemented by further techniques, they provide the ground on which the celebrated threshold theorem can then be established. These are quantum codes, error discretization and transversal encoded gates. The purpose of this article was to explain three key ideas upon which the theory of fault-tolerant quantum computation is built. Put simply, a gate error below the threshold is effectively as good as no error at all. This is the content of the threshold theorem of fault-tolerant quantum computation , which is the capstone of the theory of quantum error correction. If the effect of decoherence per operation is below a certain limit-the error threshold-then arbitrarily long and accurate quantum computation is possible. All that is needed for quantum computation are gate operations of a sufficiently high but fixed quality. But does not the extra power harnessed in quantum mechanical ways of information processing rely on the assumption that quantum operations could be performed with perfect accuracy? Would not the slightest amount of decoherence ruin any quantum computation if it only takes long enough? ‘No’ says the theory of quantum fault tolerance . Assuming, of course, that a quantum computer could actually be built. The discovery of an efficient quantum algorithm to break the widespread Rivest–Shamir–Adleman cryptosystem showed that quantum computing has enormous potential for solving hard computational problems.
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