computational complexity

, Volume 26, Issue 4, pp 765–833 | Cite as

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

  • Leslie Ann Goldberg
  • Heng Guo


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.


Counting Complexity Ising model Tutte polynomial Approximate Counting 

Subject classification

68Q17 Computational difficulty of problems 


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  1. Aaronson Scott (2005) Quantum computing, postselection, and probabilistic polynomial-time. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2063): 3473–3482CrossRefzbMATHMathSciNetGoogle Scholar
  2. Aaronson Scott, Arkhipov Alex (2013) The Computational Complexity of Linear Optics. Theory of Computing 9: 143–252CrossRefzbMATHMathSciNetGoogle Scholar
  3. Dorit Aharonov & Itai Arad (2011). The BQP-hardness of approximating the Jones polynomial. New Journal of Physics 13(3), 035 019.Google Scholar
  4. Bordewich M., Freedman M., Lovász L., Welsh D. (2005) Approximate Counting and Quantum Computation. Combin. Probab. Comput. 14(5–6): 737–754CrossRefzbMATHMathSciNetGoogle Scholar
  5. Michael J. Bremner, Richard Jozsa & Dan J. Shepherd (2011). Classical Simulation of Commuting Quantum Computations implies Collapse of the Polynomial Hierarchy. Proc. R. Soc. A 467(2126), 459–472.Google Scholar
  6. Y. Bugeaud (2004). Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics. Cambridge University Press. ISBN 9781139455671.Google Scholar
  7. Cai Jin-Yi, Lu Pinyan, Xia Mingji (2014) The complexity of complex weighted Boolean #CSP. J. Comput. Syst. Sci. 80(1): 217–236CrossRefzbMATHMathSciNetGoogle Scholar
  8. G. De las Cuevas, W. Dür, M. Van den Nest & M. A. Martin-Delgado (2011). Quantum algorithms for classical lattice models. New Journal of Physics 13(9), 093 021.Google Scholar
  9. Michael H. Freedman, Alexei Kitaev, Michael J. Larsen & Zhenghan Wang (2003). Topological quantum computation. Bull. Amer. Math. Soc. (N.S.) 40(1), 31–38.Google Scholar
  10. H. Freedman Michael, Larsen Michael, Wang Zhenghan (2002) A modular functor which is universal for quantum computation. Comm. Math. Phys. 227(3): 605–622CrossRefzbMATHMathSciNetGoogle Scholar
  11. Keisuke Fujii & Tomoyuki Morimae (2013). Quantum Commuting Circuits and Complexity of Ising Partition Functions. CoRR arXiv:1311.2128.
  12. Joseph Geraci & Daniel A Lidar (2010). Classical Ising model test for quantum circuits. New Journal of Physics 12(7), 075 026.Google Scholar
  13. Ann Goldberg Leslie, Jerrum Mark (2008) Inapproximability of the Tutte polynomial. Inf. Comput. 206(7): 908–929CrossRefzbMATHMathSciNetGoogle Scholar
  14. Ann Goldberg Leslie, Jerrum Mark (2012) Inapproximability of the Tutte polynomial of a planar graph. Computational Complexity 21(4): 605–642CrossRefzbMATHMathSciNetGoogle Scholar
  15. Ann Goldberg Leslie, Jerrum Mark (2014) The Complexity of Computing the Sign of the Tutte Polynomial. SIAM J. Comput. 43(6): 1921–1952CrossRefMathSciNetGoogle Scholar
  16. Iblisdir S., Cirio M., Kerans O., Brennen G. K. (2014) Low depth quantum circuits for Ising models. Annals of Physics 340(205): 205–251CrossRefzbMATHMathSciNetGoogle Scholar
  17. Jaeger F., Vertigan D. L., Welsh D. J. A. (1990) On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108(1): 35–53CrossRefzbMATHMathSciNetGoogle Scholar
  18. Jerrum Mark, Sinclair Alistair (1993) Polynomial-Time Approximation Algorithms for the Ising Model. SIAM J. Comput. 22(5): 1087–1116CrossRefzbMATHMathSciNetGoogle Scholar
  19. Jozsa Richard, Van den Nest Marrten (2014) Classical Simulation Complexity of Extended Clifford Circuits. Quantum Info. Comput. 14(7&8): 633–648MathSciNetGoogle Scholar
  20. Kuperberg Greg (2015) How Hard Is It to Approximate the Jones Polynomial?. Theory of Computing 11: 183–219CrossRefzbMATHMathSciNetGoogle Scholar
  21. A. Matsuo, K. Fujii & N. Imoto (2014). A quantum algorithm for additive approximation of Ising partition functions. Phys. Rev. A 90, 022 304.Google Scholar
  22. Michael A. Nielsen & Isaac L. Chuang (2004). Quantum Computation and Quantum Information (Cambridge Series on Information and the Natural Sciences). Cambridge University Press, 1st edition. ISBN 0521635039.Google Scholar
  23. Scott Provan J., O. Ball Michael (1983) The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput. 12(4): 777–788CrossRefzbMATHMathSciNetGoogle Scholar
  24. Dan Shepherd (2010). Binary Matroids and Quantum Probability Distributions. CoRR arXiv:1005.1744.
  25. Dan J. Shepherd & Michael J. Bremner (2009). Temporally Unstructured Quantum Computation. Proc. R. Soc. A 465(2105), 1413–1439.Google Scholar
  26. Alan D. Sokal (2005). The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In Surveys in combinatorics 2005, volume 327 of London Math. Soc. Lecture Note Ser., 173–226. Cambridge Univ. Press, Cambridge.Google Scholar
  27. B. Thistlethwaite Morwen (1987) A spanning tree expansion of the Jones polynomial. Topology 26(3): 297–309CrossRefzbMATHMathSciNetGoogle Scholar
  28. G. Valiant Leslie, V. Vazirani Vijay (1986) NP is as Easy as Detecting Unique Solutions. Theor. Comput. Sci. 47(3): 85–93CrossRefzbMATHMathSciNetGoogle Scholar
  29. Ziv Abraham (1982) Relative distance—an error measure in round-off error analysis. Math. Comp. 39(160): 563–569zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK
  2. 2.School of InformaticsUniversity of EdinburghEdinburghUK

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