The question asked is "Is there a way of recovering from the errors caused by [quantum] noise in an effective way?" and Peter Shor's answer admirably covers one effective way to answer this question, namely, by designing fault-tolerant quantum computers.
An alternative effective way is very commonly encountered in engineering practice. We reason "If the noise is sufficiently large that no quantum computation is feasible, then perhaps the system dynamics can be simulated with classical resources in P."
In other words, oftentimes we can "recover in an effective way" from noise by recognizing that the noise is providing an important service to us, by exponentially reducing the computational complexity of simulating both classical and quantum systems.
The literature on noise-centric approaches to dynamical simulation is large and growing; a recent reference whose theorems are both physically motivated and pleasingly rigorous, and which includes many references to the broader literature, is Plenio and Virmani's Upper bounds on fault tolerance thresholds of noisy Clifford-based quantum computers (arXiv:0810.4340v1).
Classical dynamicists use a very different language in which noise mechanisms go by the technical name of thermostats; Frenkel and Smit's Understanding Molecular Simulation: from Algorithms to Applications (1996) provides a basic mathematical introduction.
When we transcribe classical and quantum thermostats into the language of geometric dynamics, we find (unsurprisingly) that classical and quantum methods for exploiting noise to boost simulation efficiency are essentially identical; that their respective literatures so infrequently reference one another is largely an accident of history that has been sustained by notational obstructions.
Less rigorously but more generally, the above results illuminate the origins in quantum information theory of a heuristic rule that is widely embraced by chemists, physicists, and biologists, that any classical or quantum system that is in dynamical contact with a thermal bath is likely to prove simulable with computational resources in P for all practical purposes (FAPP).
The exceptions to this heuristic, both classical and quantum, represent important open problems. Their number strikingly diminishes year-by-year; the biennial Critical Assessment of Structure Prediction (CASP) provides one objective measure of this improvement.
The fundamental limits to this noise-driven, many-decade "More than Moore" progress in simulation capability are at present imperfectly known. Needless to say, in the long run our steadily improving understanding of these limits will bring us nearer to building quantum computers, while in the short run, this knowledge greatly assists us in efficiently simulating systems that are not quantum computers. Either way, it's good news.