  We analyze the long time behavior of a quantum computer running a quantum
error correction (QEC) code in the presence of a correlated environment.
Starting from a Hamiltonian formulation of realistic noise models, and assuming
that QEC is indeed possible, we find formal expressions for the probability of
a faulty path and the residual decoherence encoded in the reduced density
matrix. Systems with non-zero gate times (``long gates'') are included in our
analysis by using an upper bound on the noise. In order to introduce the local
error probability for a qubit, we assume that propagation of signals through
the environment is slower than the QEC period (hypercube assumption). This
allows an explicit calculation in the case of a generalized spin-boson model
and a quantum frustration model. The key result is a dimensional criterion: If
the correlations decay sufficiently fast, the system evolves toward a
stochastic error model for which the threshold theorem of fault-tolerant
quantum computation has been proven. On the other hand, if the correlations
decay slowly, the traditional proof of this threshold theorem does not hold.
This dimensional criterion bears many similarities to criteria that occur in
the theory of quantum phase transitions.
