  Is Fully Polynomial-time Randomized Approximation Scheme (FPRAS) for a
problem via an MCMC algorithm possible when it is known that rapid mixing
provably fails? We introduce several weight-preserving maps for the
eight-vertex model on planar and on bipartite graphs, respectively. Some are
one-to-one, while others are holographic which map superpositions of
exponentially many states from one setting to another, in a quantum-like
many-to-many fashion. In fact we introduce a set of such mappings that forms a
group in each case. Using some holographic maps and their compositions we
obtain FPRAS for the eight-vertex model at parameter settings where it is known
that rapid mixing provably fails due to an intrinsic barrier. This FPRAS is
indeed the same MCMC algorithm, except its state space corresponds to
superpositions of the given states, where rapid mixing holds. FPRAS is also
given for torus graphs for parameter settings where natural Markov chains are
known to mix torpidly. Our results show that the eight-vertex model is the
first problem with the provable property that while NP-hard to approximate on
general graphs (even #P-hard for planar graphs in exact complexity), it
possesses FPRAS on both bipartite graphs and planar graphs in substantial
regions of its parameter space.
