The Complexity of Approximating complex-valued Ising and Tutte partition functions

Abstract

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.

Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

“…As with the case of Boson Sampling, the worst-case hardness of multiplicative approximations to both Z(ω) and ngap(f ) ( [11] and Appendix D) implies via standard results on random-self-reducibility [13] that there exists some distribution over the choices of these functions that is #P-hard on average -but not necessarily the distributions that we require for Conjectures 2 and 3.…”

“…See Appendix C for a description of this construction. A consequence is that classical computation of such amplitudes is #P-hard, even up to constant multiplicative error [3,11] or exponentially small additive error [9].…”

“…It is known that the family of partition functions Z(ω) parametrised as above is #P-hard to compute in the worst case up to the above multiplicative error bound [10,11]. Conjecture 2 thus states that this worst-case hardness result can be improved to an average-case hardness result.…”

“…See Appendix C for a description of this construction. A consequence is that classical computation of such amplitudes is #P-hard, even up to constant multiplicative error [3,11] or exponentially small additive error [9].Approximation of general IQP circuits -We first prove a key technical ingredient, which relates approximate sampling from the output distributions of IQP circuits to approximating individual output probabilities. This is essentially the same argument as used in [4] for the permanent, although we believe it becomes substantially simpler in the setting of IQP.…”

“…where the exponentiated sum is over the complete graph on n vertices, w ij and v k are real edge and vertex weights, and ω ∈ C. Then, for any ω = e iθ , Z(ω) arises straightforwardly as an amplitude of some IQP circuit C I (ω): 0| ⊗n C I (ω)|0 ⊗n = Z(ω)/2 n (see Appendix A and [7][8][9][10][11]). For our purposes it is sufficient to restrict to the case where ω = e iπ/8 and the weights are picked by choosing uniformly at random from the set {0, .…”