The complexity of approximating the complex-valued Potts model
We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work i...
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| Published in | Computational complexity Vol. 31; no. 1 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
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Springer International Publishing
01.06.2022
Springer Nature B.V |
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| ISSN | 1016-3328 1420-8954 1420-8954 |
| DOI | 10.1007/s00037-021-00218-x |
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| Abstract | We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q
≥
2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations. |
|---|---|
| AbstractList | We study the complexity of approximating the partition function of
the q-state Potts model and the closely related Tutte polynomial
for complex values of the underlying parameters. Apart from the
classical connections with quantum computing and phase transitions
in statistical physics, recent work in approximate counting has
shown that the behaviour in the complex plane, and more precisely
the location of zeros, is strongly connected with the complexity of
the approximation problem, even for positive real-valued parameters.
Previous work in the complex plane by Goldberg and Guo focused on
q = 2, which corresponds to the case of the Ising model; for q > 2,
the behaviour in the complex plane is not as well understood and
most work applies only to the real-valued Tutte plane. Our main
result is a complete classification of the complexity of the
approximation problems for all non-real values of the parameters, by
establishing #P-hardness results that apply even when restricted to
planar graphs. Our techniques apply to all q
$$\geq$$
≥
2 and further
complement/refine previous results both for the Ising model and the
Tutte plane, answering in particular a question raised by Bordewich,
Freedman, Lovász and Welsh in the context of quantum
computations. We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations. We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations. |
| ArticleNumber | 2 |
| Author | Galanis, Andreas Herrera-Poyatos, Andrés Goldberg, Leslie Ann |
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| Cites_doi | 10.1137/12088330X 10.1145/3337785 10.1145/3418056 10.4086/toc.2015.v011a006 10.1090/S0025-5718-1988-0917831-4 10.1137/0212053 10.1007/978-3-319-51829-9 10.1006/jctb.2001.2057 10.1145/2371656.2371660 10.1109/FOCS.2019.00085 10.1017/S0963548305007005 10.1017/CBO9780511734885.009 10.1007/978-3-540-85521-7_4 10.1112/jlms.12286 10.1016/0040-9383(87)90003-6 10.1007/BF01213009 10.1145/3357713.3384322 10.1007/3-540-51084-2_22 10.1137/16M1101003 10.1063/1.5082552 10.1137/1.9780898719796 10.1007/s00037-012-0046-4 10.1103/PhysRev.87.404 10.1017/CBO9780511755316 10.1307/mmj/1541667626 10.1017/S0963548300000705 10.1137/S0097539704446797 10.1017/CBO9780511752506 10.1142/S0217979205032759 10.1137/18M1184485 10.1016/j.ic.2008.04.003 10.1137/1.9781611975994.11 10.1017/S0305004100027419 10.1137/0222066 10.1007/978-1-4684-6802-1 10.1007/BF01877590 10.1017/S0305004100068936 10.1007/s10955-018-2199-2 10.1006/jsco.1997.0158 10.1145/3448645 10.1007/s00037-017-0162-2 10.1017/S0963548319000105 |
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| Keywords | Counting Complexity Ising model Tutte polynomial 68Q17 Computational difficulty of problems Potts model Approximate Counting |
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| References_xml | – reference: O. J. Heilmann & E. H. Lieb (1972). Theory of monomer-dimer systems. Communications in Mathematical Physics25(3), 190–232. – reference: A. W. Strzeboński (1997). Computing in the field of complex algebraic numbers. Journal of Symbolic Computation24(6), 647–656. – reference: R. J. Bradford & J. H. Davenport (1989). Effective tests for cyclotomic polynomials. In Symbolic and algebraic computation, volume 358 of Lecture Notes in Comput. Sci., 244–251. Springer, Berlin. – reference: M. Waldschmidt (2000). Diophantine approximation on linear algebraic groups, volume 326 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. Transcendence properties of the exponential function in several variables. – reference: J. S. Provan & M. O. Ball (1983). The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing12(4), 777–788. – reference: L. A. Goldberg & M. Jerrum (2019). Approximating Pairwise Correlations in the IsingModel. ACM Transactions on Computation Theory11(4), 23. – reference: M. Bordewich, M. Freedman, L. Lovász & D. Welsh (2005). Approximate counting and quantum computation. Combinatorics, Probability and Computing14(5-6), 737–754. – reference: K.-I Ko (1991). Complexity theory of real functions. Progress in Theoretical Computer Science. Birkh¨auser Boston, Inc., Boston, MA. – reference: E. M. Stein & R. Shakarchi (2010). Complex analysis, volume 2. Princeton University Press. – reference: L. A. Goldberg & M. Jerrum (2012b). Inapproximability of the Tutte polynomial of a planar graph. Computational Complexity21(4), 605–642. – reference: R. B. Potts (1952). Some generalized order-disorder transformations. Proceedings of the Cambridge Philosophical Society48, 106–109. – reference: H. Peters & G. Regts (2019). On a Conjecture of Sokal Concerning Roots of the Independence Polynomial. Michigan Mathematical Journal68(1), 33–55. – reference: I. Stewart (2004). Galois Theory. Chapman & Hall/CRC Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 3rd edition. – reference: J. Liu, A. Sinclair & P. Srivastava (2019). A Deterministic Algorithm for Counting Colorings with 2-Delta Colors. In IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS 2019), 1380– 1404. – reference: D. J. A. Welsh (1993). Complexity: knots, colourings and counting, volume 186 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge. – reference: Bill Jackson (1993). A zero-free interval for chromatic polynomials of graphs. Combinatorics, Probability and Computing2(3), 325–336. ISSN 0963-5483. URL https://doi.org/10.1017/S0963548300000705. – reference: J. Liu, A. Sinclair & P. Srivastava (2019a). Fisher zeros and correlation decay in the Ising model. Journal of Mathematical Physics60(10), 103 304. – reference: M. Brin & G. Stuck (2002). 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| Snippet | We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the... We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Approximation Complexity Computational Mathematics and Numerical Analysis Computer Science Ising model Mathematical models Parameters Partitions (mathematics) Phase transitions Polynomials Quantum computing |
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| Title | The complexity of approximating the complex-valued Potts model |
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