Advertisement

Journal of Statistical Physics

, Volume 160, Issue 5, pp 1389–1404 | Cite as

Tensor Network Contractions for #SAT

  • Jacob D. Biamonte
  • Jason Morton
  • Jacob Turner
Article

Abstract

The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of #SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g. 2-SAT, which is in \(\mathsf{{P}}\)), determining the number of solutions can be #\(\mathsf{{P}}\)-hard. Recently, computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics-inspired techniques. By these methods, we give an algorithm using an axiomatic tensor contraction language for n-variable #SAT instances with complexity \(O((g+cd)^{O(1)} 2^c)\) where c is the number of COPY-tensors, g is the number of gates, and d is the maximal degree of any COPY-tensor. Thus, n-variable counting problems can be solved efficiently when their tensor network expression has at most \(O(\log n)\) COPY-tensors and polynomial fan-out. This framework also admits an intuitive proof of a variant of the Tovey conjecture (the r,1-SAT instance of the Dubois–Tovey theorem). This study increases the theory, expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science.

Keywords

Statistical physics Complexity Computational physics Quantum physics 

Notes

Acknowledgments

We thank Tobias Fritz and Eduardo Mucciolo for providing feedback. JDB acknowledges financial support from the Fondazione Compagnia di San Paolo through the Q-ARACNE project and the Foundational Questions Institute (FQXi, under Grant FQXi-RFP3-1322). JM and JT acknowledges the NSF (under Grant NSF-1007808) for financial support.

References

  1. 1.
    Barahona, F.: On the computational complexity of ising spin glass models. J. Phys. A 15(10), 3241 (1982)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Monasson, Rémi, Zecchina, Riccardo, Kirkpatrick, Scott, Selman, Bart, Troyansky, Lidror: Determining computational complexity from characteristic phase transitions. Nature 400(6740), 133–137 (1999)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Goerdt, Andreas: A threshold for unsatisfiability. J. Comp. Syst. Sci. 53(3), 469–486 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kirkpatrick, Scott, Selman, Bart: Critical behavior in the satisfiability of random boolean expressions. Science 264(5163), 1297–1301 (1994)MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Nishimura, Naomi, Ragde, Prabhakar, Szeider, Stefan: Solving# sat using vertex covers. Acta Informa. 44(7–8), 509–523 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fischer, Eldar: Johann A Makowsky, and Elena V Ravve. Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete. Appl. Math. 156(4), 511–529 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Igor, L.: Markov and Yaoyun Shi. Simulating quantum computation by contracting tensor networks. SIAM J. Comput. 38(3), 963–981 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Samer, M., Szeider, S.: A fixed-parameter algorithm for #SAT with parameter incidence treewidth (2006). arXiv:cs/0610174
  10. 10.
    Bacchus, F., Dalmao, S., Pitassi, T.: Algorithms and complexity results for# sat and bayesian inference. In: Proceedings of 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 340–351 (2003)Google Scholar
  11. 11.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Verstraete, F., Murg, V., Cirac, J.I.: Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Adv. Phys. 57(2), 143–224 (2008)ADSCrossRefGoogle Scholar
  13. 13.
    Johnson, T.H., Clark, S.R., Jaksch, D.: Dynamical simulations of classical stochastic systems using matrix product states. Phys. Rev. E 82(3), 036702 (2010)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Evenbly, G., Vidal, G.: Tensor network states and geometry. J. Stat. Phys. 145, 891–918 (2011)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Eisert, J.: Entanglement and tensor network states. Model. Simul. 3, 520 (2013). http://www.cond-mat.de/events
  16. 16.
    Penrose, Roger: Applications of negative dimensional tensors. In: Welsh, D. (ed.) Combinatorial Mathematics and its Applications, pp. 221–244. Academic Press, New York (1971)Google Scholar
  17. 17.
    Griffiths, R.B.: Channel kets, entangled states, and the location of quantum information. PRA 71(4), 042337 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    Griffiths, R.B., Wu, S., Yu, L., Cohen, S.M.: Atemporal diagrams for quantum circuits. PRA 73(5), 052309 (2006)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Gross, D., Eisert, J.: Novel schemes for measurement-based quantum computation. Phys. Rev. Lett. 98(22), 220503 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    Biamonte, J.D., Clark, S.R., Jaksch, D.P.: Categorical Tensor Network States. AIP Advances 1(4), 042172 (2011). http://scitation.aip.org
  21. 21.
    Johnson, T.H., Biamonte, J.D., Clark, S.R., Jaksch, D.: Solving search problems by strongly simulating quantum circuits. Sci. Rep. 3 (2013). http://www.nature.com/srep/
  22. 22.
    Chamon, C., Mucciolo, E.R.: Virtual parallel computing and a search algorithm using matrix product states. Phys. Rev. Lett. 109(3), 030503 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    Morton, J., Turner, J.: Generalized counting constraint satisfaction problems with determinantal circuits. Linear Algebra Appl. 466, 357–381 (2015). doi: 10.1016/j.laa.2014.09.050
  24. 24.
    Kliesch, M., Gross, D., Eisert, J.: Matrix product operators and states—NP-hardness and undecidability. Phys. Rev. Lett. 113, 160503 (2014). doi: 10.1103/PhysRevLett.113.160503
  25. 25.
    Morton, J., Biamonte, J.: Undecidability in tensor network states. Phys. Rev. A 86(3), 030301(R) (2012)ADSCrossRefGoogle Scholar
  26. 26.
    Wolf, M.M., Cubitt, T.S., Perez-Garcia, D.: Are problems in quantum information theory (un) decidable? (2011). arXiv:1111.5425
  27. 27.
    Damm, C., Holzer, M., McKenzie, P.: The complexity of tensor calculus. Comput. Complex. 11(1), 54–89 (2003)MathSciNetGoogle Scholar
  28. 28.
    Eisert, J., Müller, M.P., Gogolin, C.: Quantum measurement occurrence is undecidable. Phys. Rev. Lett. 108(26), 260501 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    Chamon, C., Mucciolo, E.R.: Rényi entropies as a measure of the complexity of counting problems. J. Stat. Mech. 4, 8 (2013)Google Scholar
  30. 30.
    Biamonte, J.D.: Nonperturbative k -body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins. PRA 77(5), 052331 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Denny, S.J., Biamonte, J.D., Jaksch, D., Clark, S.R.: Algebraically contractible topological tensor network states. J. Phys. A 45(1), 015309 (2012)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Whitfield, J.D., Faccin, M., Biamonte, J.D.: Ground-state spin logic. EPL (Europhys. Lett.) 99, 57004 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    Alsina, D., Latorre, J.I.: Tensor Networks for Frustrated Systems: Emergence of Order from Simplex Entanglement. ArXiv e-prints, December 2013Google Scholar
  34. 34.
    Gharibian, S., Landau, Z., Shin, S.W., Wang, G.: Tensor network non-zero testing. ArXiv e-prints, June 2014Google Scholar
  35. 35.
    Valiant, Leslie G.: The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    García-Sáez, Artur, Latorre, José I.: An exact tensor network for the 3SAT problem. Quantum Inform. Comput. 12(3–4), 283–292 (2012)Google Scholar
  37. 37.
    Kempe, Julia, Kitaev, Alexei, Regev, Oded: The complexity of the local hamiltonian problem. SIAM J. Comput. 35(5), 1070–1097 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Penrose, R.: The theory of quantized directions. Unpublished (1967)Google Scholar
  39. 39.
    Lafont, Yves: Towards an algebraic theory of boolean circuits. J. Pure Appl. Algebr. 184, 2003 (2003)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Tovey, Craig A.: A simplified NP-complete satisfiability problem. Discrete. Appl. Math. 8(1), 85–89 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Dubois, Olivier: On the r, s-SAT satisfiability problem and a conjecture of tovey. Discrete. Appl. Math. 26(1), 51–60 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Halin, Rudolf: S-functions for graphs. J. Geom. 8(1–2), 171–186 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lauritzen, S.L., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. J. R. Stat. Soc. B 50, 157–224 (1988)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Xia, Mingji, Zhang, Peng, Zhao, Wenbo: Computational complexity of counting problems on 3-regular planar graphs. Theor. Comput. Sci. 384(1), 111–125 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Rényi, A.:. On measures of entropy and information. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–561 (1961)Google Scholar
  46. 46.
    Biamonte, J., Bergholm, V., Lanzagorta, M.: Tensor network methods for invariant theory. J. Phys. A Math. Gener. 46, 5301 (2013)MathSciNetADSGoogle Scholar
  47. 47.
    Baez, J.C.: Renyi entropy and free energy (2011). arXiv:1102.2098
  48. 48.
    Valiant, Leslie G., Vazirani, Vijay V.: NP is as easy as detecting unique solutions. Theor. Compu. Sci. 47, 85–93 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Bravyi, Sergey: Contraction of matchgate tensor networks on non-planar graphs. Contemp. Math. 482, 179–211 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ISI FoundationTorinoItaly
  2. 2.Department of MathematicsThe Pennsylvania State UniversityState CollegeUSA

Personalised recommendations