The complexity of approximating complex-valued Ising and Tutte partition functions with applications to quantum simulation

We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters. Our results are motivated by the study of the quantum complexity classes BQP and IQP. Recent results show how to encode quantum computations as evaluations of 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. The main motivation for this paper is to go the other way around. Partition functions are combinatorial in nature and classifying the difficulty of approximating these partition functions 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 of the partition function for complex edge interactions. We also give the first classification of the complexity of additively approximating the argument of the partition function (and of approximating the partition function according to a natural complex metric). Using our classifications, we then revisit the connections to quantum computation, drawing conclusions that are different from (and incomparable to) the results in the quantum complexity literature. We show that strong simulation of IQP within any constant factor is #P-hard, even for the restricted class of circuits studied by Bremner et al. We also show that computing the sign of the Tutte polynomial is #P-hard at a certain point related to the simulation of BQP. Finally, motivated by further BQP-hardness results, we study the problem of approximating the norm of the Ising partition function in the presence of external fields. We give a complete dichotomy for the case in which 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.