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We study the complexity of approximately evaluating the Ising and Tuttepartition functions with complex parameters. Our results are partly motivatedby the study of the quantum complexity classes BQP and IQP. Recent results showhow to encode quantum computations as evaluations of classical partitionfunctions. These results rely on interesting and deep results about quantumcomputation 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 complexityof (classically) approximating the Ising and Tutte partition functions withcomplex parameters. Partition functions are combinatorial in nature andquantifying their approximation complexity does not require a detailedunderstanding of quantum computation. Using combinatorial arguments, we givethe first full classification of the complexity of multiplicativelyapproximating the norm and additively approximating the argument of the Isingpartition function for complex edge interactions (as well as of approximatingthe partition function according to a natural complex metric). We also studythe norm approximation problem in the presence of external fields, for which wegive a complete dichotomy when the parameters are roots of unity. Previousresults were known just for a few such points, and we strengthen these resultsfrom BQP-hardness to #P-hardness. Moreover, we show that computing the sign ofthe Tutte polynomial is #P-hard at certain points related to the simulation ofBQP. Using our classifications, we then revisit the connections to quantumcomputation, drawing conclusions that are a little different from (andincomparable to) ones in the quantum literature, but along similar lines.

The complexity of approximating complex-valued Ising and Tutte partition functions