article

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

Authors:

Leslie Ann Goldberg,

Heng GuoAuthors Info & Claims

Computational Complexity, Volume 26, Issue 4

Pages 765 - 833

https://doi.org/10.1007/s00037-017-0162-2

Published: 01 December 2017 Publication History

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 in (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.

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Cited By

Buys PGalanis APatel VRegts GMarx D(2021)Lee-yang zeros and the complexity of the ferromagnetic ising model on bounded-degree graphsProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458155(1508-1519)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458155
Bezáková IGalanis AGoldberg LŠtefankovič D(2021)The Complexity of Approximating the Matching Polynomial in the Complex PlaneACM Transactions on Computation Theory10.1145/344864513:2(1-37)Online publication date: 19-Apr-2021
https://dl.acm.org/doi/10.1145/3448645
Bezáková IGalanis AGoldberg LŠtefankovič DDiakonikolas IKempe DHenzinger M(2018)Inapproximability of the independent set polynomial in the complex planeProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188788(1234-1240)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188788

The Complexity of Approximating complex-valued Ising and Tutte partition functions
1. Theory of computation

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cover image Computational Complexity

Computational Complexity Volume 26, Issue 4

December 2017

226 pages

ISSN:1016-3328

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Copyright © Copyright © 2017 Springer International Publishing AG, part of Springer Nature.

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Birkhauser Verlag

Switzerland

Publication History

Published: 01 December 2017

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Cited By

Buys PGalanis APatel VRegts GMarx D(2021)Lee-yang zeros and the complexity of the ferromagnetic ising model on bounded-degree graphsProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458155(1508-1519)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458155
Bezáková IGalanis AGoldberg LŠtefankovič D(2021)The Complexity of Approximating the Matching Polynomial in the Complex PlaneACM Transactions on Computation Theory10.1145/344864513:2(1-37)Online publication date: 19-Apr-2021
https://dl.acm.org/doi/10.1145/3448645
Bezáková IGalanis AGoldberg LŠtefankovič DDiakonikolas IKempe DHenzinger M(2018)Inapproximability of the independent set polynomial in the complex planeProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188788(1234-1240)Online publication date: 20-Jun-2018
https://dl.acm.org/doi/10.1145/3188745.3188788

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