  Given a quantum channel and a state which satisfy a fixed point equation
approximately (say, up to an error $\varepsilon$), can one find a new channel
and a state, which are respectively close to the original ones, such that they
satisfy an exact fixed point equation? It is interesting to ask this question
for different choices of constraints on the structures of the original channel
and state, and requiring that these are also satisfied by the new channel and
state. We affirmatively answer the above question, under fairly general
assumptions on these structures, through a compactness argument. Additionally,
for channels and states satisfying certain specific structures, we find
explicit upper bounds on the distances between the pairs of channels (and
states) in question. When these distances decay quickly (in a particular,
desirable manner) as $\varepsilon\to 0$, we say that the original approximate
fixed point equation is rapidly fixable. We establish rapid fixability, not
only for general quantum channels, but also when the original and new channels
are both required to be unitary, mixed unitary or unital. In contrast, for the
case of bipartite quantum systems with channels acting trivially on one
subsystem, we prove that approximate fixed point equations are not rapidly
fixable. In this case, the distance to the closest channel (and state) which
satisfy an exact fixed point equation can depend on the dimension of the
quantum system in an undesirable way. We apply our results on approximate fixed
point equations to the question of robustness of quantum Markov chains (QMC)
and establish the following: For any tripartite quantum state, there exists a
dimension-dependent upper bound on its distance to the set of QMCs, which
decays to zero as the conditional mutual information of the state vanishes.
