Hardness of Approximation for Quantum Problems

  • Sevag Gharibian
  • Julia Kempe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Our work thus yields the first known hardness of approximation results for a quantum complexity class. Using these techniques, we also obtain hardness of approximation for the class QCMA. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy (Umans 1999). We close by showing that a variant of the local Hamiltonian problem with hybrid classical-quantum ground states is complete for the second level of our quantum hierarchy.

Keywords

Quantum Circuit Quantum Problem Quantum Setting Polynomial Hierarchy Quantum Generalization 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AALV09]
    Aharonov, D., Arad, I., Landau, Z., Vazirani, U.: The detectibility lemma and quantum gap amplification. In: 41st ACM Syposium on Theory of Computing, vol. 287, pp. 417–426 (2009)Google Scholar
  2. [Aar06]
    Aaronson, S.: The quantum PCP manifesto (2006), http://scottaaronson.com/blog/?p=139
  3. [AN02]
    Aharonov, D., Naveh, T.: Quantum NP - A survey. Preprint at arXiv:quant-ph/0210077v1 (2002)Google Scholar
  4. [Ara10]
    Arad, I.: A note about a partial no-go theorem for quantum PCP. Preprint at arXiv:quant-ph/1012.3319 (2010)Google Scholar
  5. [FLvMV05]
    Fortnow, L., Lipton, R., van Melkebeek, D., Viglas, A.: Time-space lower bounds for satisfiability. Journal of the ACM 52, 835–865 (2005)MathSciNetCrossRefGoogle Scholar
  6. [For00]
    Fortnow, L.: Time-space tradeoffs for satisfiability. Journal of Computer and System Sciences 60(2), 337–353 (2000)MathSciNetMATHCrossRefGoogle Scholar
  7. [Has12]
    Hastings, M.B.: Trivial low energy states for commuting hamiltonians, and the quantum PCP conjecture. Preprint at arXiv:quant-ph/1201.3387 (2012)Google Scholar
  8. [Hem02]
    Hemaspaandra, L.: SIGACT news complexity theory column 38. ACM SIGACT News 33(4) (2002); Guest column by Schaefer, M., Umans, C.Google Scholar
  9. [KKR06]
    Kempe, J., Kitaev, A., Regev, O.: The complexity of the local Hamiltonian problem. SIAM Journal on Computing 35(5), 1070–1097 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. [KR03]
    Kempe, J., Regev, O.: 3-local Hamiltonian is QMA-complete. Quantum Information & Computation 3(3), 258–264 (2003)MathSciNetMATHGoogle Scholar
  11. [KSV02]
    Kitaev, A., Shen, A., Vyalyi, M.: Classical and Quantum Computation. American Mathematical Society (2002)Google Scholar
  12. [Lau83]
    Lautemann, C.: BPP and the polynomial time hierarchy. Information Processing Letters 17, 215–218 (1983)MathSciNetMATHCrossRefGoogle Scholar
  13. [Sip83]
    Sipser, M.: A complexity theoretic approach to randomness. In: 15th Symposium on Theory of Computing, pp. 330–335. ACM Press (1983)Google Scholar
  14. [SZ94]
    Srinivasan, A., Zuckerman, D.: Computing with very weak random sources. In: 35th Symposium on Foundations of Computer Science, pp. 264–275 (1994)Google Scholar
  15. [TSUZ07]
    Ta-Shma, A., Umans, C., Zuckerman, D.: Lossless condensers, unbalanced expanders, and extractors. Combinatorica 27(2), 213–240 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. [Uma99]
    Umans, C.: Hardness of approximating \(\Sigma_2^p\) minimization problems. In: 40th Symposium on Foundations of Computer Science, pp. 465–474 (1999)Google Scholar
  17. [Wat09]
    Watrous, J.: Quantum computational complexity. In: Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science, ch. 17, pp. 7174–7201. Springer (2009)Google Scholar
  18. [Yam02]
    Yamakami, T.: Quantum NP and a quantum hierarchy. In: 2nd IFIP International Conference on Theoretical Computer Science, pp. 323–336. Kluwer Academic Publishers (2002)Google Scholar
  19. [Zuc96]
    Zuckerman, D.: On unapproximable versions of NP-complete problems. SIAM Journal on Computing 25(6), 1293–1304 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sevag Gharibian
    • 1
  • Julia Kempe
    • 2
    • 3
  1. 1.David R. Cheriton School of Computer Science and Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  2. 2.CNRS & LIAFA, University Paris Diderot - Paris 7ParisFrance
  3. 3.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael

Personalised recommendations