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 BQPhardness to #Phardness. Moreover, we show that computing the sign of the Tutte polynomial is #Phard 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 2 Goldberg & Guo quantum literature, but along similar lines.
Original language  English 

Pages (fromto)  765–833 
Number of pages  70 
Journal  Computational complexity 
Volume  26 
Issue number  4 
DOIs  
Publication status  Published  13 Sep 2017 
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Profiles

Heng Guo
 School of Informatics  Lecturer in Algorithms and Complexity
 Laboratory for Foundations of Computer Science  Lecturer in Algorithms and Complexity
 Foundations of Computation
Person: Academic: Research Active (Teaching)