  The construction of large, coherent quantum systems necessary for quantum
computation remains an entreating but elusive goal, due to the ubiquitous
nature of decoherence. Recent progress in quantum error correction schemes have
given new hope to this field, but thus far, the codes presented in the
literature assume a restricted number of errors and error free encoding,
decoding, and measurement. We investigate a specific scenario without these
assumptions; in particular, we evaluate a scheme to preserve a single quantum
bit against phase damping using a three-qubit encoding based on Shor. By
applying a new formalism which gives simple operators for decoherence and noisy
logic gates, we find the fidelity of the stored qubit as a function of time,
including decoherence which occurs not only during storage but also during
processing. We generalize our results to include any source of error, and
derive an upper limit on the allowable decoherence per timestep. Physically,
our results suggest the feasibility of engineering artificial metastable states
through repeated error correction.
