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Statistical Mechanics of Classical and Quantum Computational Complexity

  • C. R. Laumann
  • R. Moessner
  • A. Scardicchio
  • S. L. Sondhi
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 843)

Abstract

The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (“NP-complete”) and those which are likely hard for quantum computers (“QMA-complete”) by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances—typically those that involve optimization—by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness.

Keywords

Quantum Computer Ground State Energy Complexity Theory Interaction Graph Satisfying Assignment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

We very gratefully acknowledge collaborations with Andreas Läuchli, in particular on the work reported in Ref. [34]. Chris Laumann was partially supported by a travel award of ICAM-I2CAM under NSF grant DMR-0844115.

References

  1. 1.
    Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15, 3241 (1982)Google Scholar
  2. 2.
    Aharonov, D., Gottesman, D., Kempe, J.: The power of quantum systems on a line. commun. Math. Phys. 287, 41 (2009)CrossRefzbMATHADSMathSciNetGoogle Scholar
  3. 3.
    Aaronson, S.: Guest column: NP-complete problems and physical reality. SIGACT News. 36, 30–52 (2005)CrossRefGoogle Scholar
  4. 4.
    Arora, S., Barak, B.: Complexity Theory: A Modern Approach. Cambridge University Press, Cambridge, MA (2009)zbMATHGoogle Scholar
  5. 5.
    Aharonov, D., and Naveh, T.: Quantum NP—A Survey, arXiv:quant-ph/0210077v1Google Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences. W. H. Freeman & Co Ltd, San Francisco, CA (1979)Google Scholar
  7. 7.
    Bravyi, S.: Efficient algorithm for a quantum analogue of 2-SAT, arXiv:quant-ph/0602108v1Google Scholar
  8. 8.
    Laumann, C.R., Moessner, R., Scardicchio, A., Sondhi, S.L.: Phase transitions and random quantum satisfiability. Quant. Inf. Comp. 10, 0001 (2010)MathSciNetGoogle Scholar
  9. 9.
    Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., Zdeborova, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Nat. Acad. Sci. USA 104, 10318 (2007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  10. 10.
    Dubois, O.: Upper bounds on the satisfiability threshold. Theor. Comput. Sci. 265, 187 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hartmann, A.K., Weigt, M.: Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics. Wiley, Weinheim (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Braunstein, A., Mezard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability. Rand. Struct. Alg. 27, 201–226 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Mézard, M.: Physics/computer science: passing messages between disciplines. Science 301, 1685 (2003)CrossRefGoogle Scholar
  14. 14.
    Mézard, M., Zecchina, R.: Random K-satisfiability problem: from an analytic solution to an efficient algorithm. Phys. Rev. E 66, 056126 (2002)CrossRefADSGoogle Scholar
  15. 15.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812 (2002)CrossRefADSGoogle Scholar
  16. 16.
    Mézard, M., Montanari, A.: Information, Physics and Computation. Oxford University Press Inc, New York (2009)zbMATHGoogle Scholar
  17. 17.
    Laumann, C.R., Scardicchio, A., Sondhi, S.L.: Cavity method for quantum spin glasses on the Bethe lattice. Phys. Rev. B 78, 134424 (2008)CrossRefADSGoogle Scholar
  18. 18.
    Hastings, M.B.: Quantum belief propagation: an algorithm for thermal quantum systems. Phys. Rev. B 76(20), 201102–201104 (2007)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Leifer, M., Poulin, D.: Quantum graphical models and belief propagation. Ann. Phys. 323, 1899 (2008)CrossRefzbMATHADSMathSciNetGoogle Scholar
  20. 20.
    Semerjian, G., Tarzia, M., Zamponi, F.: Exact solution of the Bose-Hubbard model on the Bethe lattice. Phys. Rev. B 80, 014524 (2009)CrossRefADSGoogle Scholar
  21. 21.
    Krzakala, F., Rosso, A., Semerjian, G., Zamponi, G.: Path-integral representation for quantum spin models: application to the quantum cavity method and Monte Carlo simulations. Phys. Rev. B 78, 134428 (2008)CrossRefADSGoogle Scholar
  22. 22.
    Carleo, G., Tarzia, M., Zamponi, F.: Bose-Einstein condensation in quantum glasses. Phys. Rev. Lett. 103, 215302 (2009)CrossRefADSGoogle Scholar
  23. 23.
    Laumann, C.R., Parameswaran, S.A., Sondhi, S.L., Zamponi, F.: AKLT models with quantum spin glass ground states. Phys. Rev. B 81, 174204 (2010)CrossRefADSGoogle Scholar
  24. 24.
    Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472 (2001)CrossRefzbMATHADSMathSciNetGoogle Scholar
  25. 25.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. Siam. Rev. 50, 755 (2008)CrossRefzbMATHADSMathSciNetGoogle Scholar
  26. 26.
    Landau, L.: Zur theorie der Energieubertragung II. Phys. Sov. Union 2, 46 (1932)zbMATHGoogle Scholar
  27. 27.
    Zener, C.: Non-adiabatic crossing of energy levels. Proc. Roy. Soc. London: Ser. A 137, 696 (1932)CrossRefADSGoogle Scholar
  28. 28.
    Smelyanskiy, V., Knysh, S., Morris, R.: Quantum adiabatic optimization and combinatorial landscapes. Phys. Rev. E 70, 036702 (2004)CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Young, A.P., Knysh, S., Smelyanskiy, V.N.: Size dependence of the minimum excitation gap in the quantum adiabatic algorithm. Phys. Rev. Lett. 101, 170503 (2008)CrossRefADSGoogle Scholar
  30. 30.
    Young, A.P., Knysh, S., Smelyanskiy, V.N.: First-order phase transition in the quantum adiabatic algorithm. Phys. Rev. Lett. 104, 020502 (2010)CrossRefADSGoogle Scholar
  31. 31.
    Altshuler, B., Krovi, H., and Roland, J.: Adiabatic quantum optimization fails for random instances of NP-complete problems, arXiv:0908.2782v2Google Scholar
  32. 32.
    Altshuler, B., Krovi, H., Roland, J.: Anderson localization makes adiabatic quantum optimization fail. PNAS 107, 12446 (2010)CrossRefzbMATHADSGoogle Scholar
  33. 33.
    Movassagh, R., Farhi, E., Goldstone, J., Nagaj, D., Osborne, T.J., Shor, P.W.: Unfrustrated qudit chains and their ground states. Phys. Rev. A 82, 012318 (2010)CrossRefADSGoogle Scholar
  34. 34.
    Laumann, C.R., Läuchli, A.M., Moessner, R., Scardicchio, A., Sondhi, S.L.: Product, generic, and random generic quantum satisfiability. Phys. Rev. A 81, 062345 (2010)CrossRefADSGoogle Scholar
  35. 35.
    Bravyi, S., Moore, C., Russell A.: Bounds on the quantum satisfibility threshold, arXiv:0907.1297v2Google Scholar
  36. 36.
    Ambainis, A., Kempe, J., Sattath, O.: in 42nd Annual ACM Symposium on Theory of Computing (2009)Google Scholar
  37. 37.
    Mézard, M., Ricci-Tersenghi, F., Zecchina, R.: Two solutions to diluted p-spin models and XORSAT problems. J. Stat. Phys. 111, 505 (2003)CrossRefzbMATHGoogle Scholar
  38. 38.
    Erdös, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. Infinite finite sets 2, 609 (1975)Google Scholar
  39. 39.
    Moser, R.A., Tardos, G.: A constructive proof of the general Lovász local lemma. J. ACM. 57, 1 (2010)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Arad, I., Cubitt, T., Kempe, J., Sattath, O., Schwarz, M., Verstraete F.: Private communication (2010)Google Scholar
  41. 41.
    Govenius, J.: Junior Paper: Running Time Scaling of a 2-QSAT Adiabatic Evolution Algorithm. Princeton Junior Paper, Princeton (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • C. R. Laumann
    • 1
  • R. Moessner
    • 2
  • A. Scardicchio
    • 3
  • S. L. Sondhi
    • 1
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Max-Planck-Institut für Physik komplexer SystemeDresdenGermany
  3. 3.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

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