  We study the complexity of approximately evaluating the Ising and Tutte
partition functions with complex parameters. Our results are partly motivated
by the study of the quantum complexity classes BQP and IQP. Recent results show
how to encode quantum computations as evaluations of classical partition
functions. These results rely on interesting and deep results about quantum
computation in order to obtain hardness results about the difficulty of
(classically) evaluating the partition functions for certain fixed parameters.
  The motivation for this paper is to study more comprehensively the complexity
of (classically) approximating the Ising and Tutte partition functions with
complex parameters. Partition functions are combinatorial in nature and
quantifying their approximation complexity does not require a detailed
understanding of quantum computation. Using combinatorial arguments, we give
the first full classification of the complexity of multiplicatively
approximating the norm and additively approximating the argument of the Ising
partition function for complex edge interactions (as well as of approximating
the partition function according to a natural complex metric). We also study
the norm approximation problem in the presence of external fields, for which we
give a complete dichotomy when the parameters are roots of unity. Previous
results were known just for a few such points, and we strengthen these results
from BQP-hardness to #P-hardness. Moreover, we show that computing the sign of
the Tutte polynomial is #P-hard at certain points related to the simulation of
BQP. Using our classifications, we then revisit the connections to quantum
computation, drawing conclusions that are a little different from (and
incomparable to) ones in the quantum literature, but along similar lines.
