  We initiate a study of the classification of approximation complexity of the
eight-vertex model defined over 4-regular graphs. The eight-vertex model,
together with its special case the six-vertex model, is one of the most
extensively studied models in statistical physics, and can be stated as a
problem of counting weighted orientations in graph theory. Our result concerns
the approximability of the partition function on all 4-regular graphs,
classified according to the parameters of the model. Our complexity results
conform to the phase transition phenomenon from physics.
  We introduce a quantum decomposition of the eight-vertex model and prove a
set of closure properties in various regions of the parameter space.
Furthermore, we show that there are extra closure properties on 4-regular
planar graphs. These regions of the parameter space are concordant with the
phase transition threshold. Using these closure properties, we derive
polynomial time approximation algorithms via Markov chain Monte Carlo. We also
show that the eight-vertex model is NP-hard to approximate on the other side of
the phase transition threshold.
