Quantum Information Processing

, Volume 10, Issue 1, pp 33–52

A study of heuristic guesses for adiabatic quantum computation

  • Alejandro Perdomo-Ortiz
  • Salvador E. Venegas-Andraca
  • Alán Aspuru-Guzik
Article

Abstract

Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.

Keywords

Heuristic algorithms Adiabatic quantum computation Adiabatic preparation Quantum algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution, quant-ph/0001106 (2000)Google Scholar
  2. 2.
    Childs A.M., Farhi E., Preskill J.: Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (2001)CrossRefADSGoogle Scholar
  3. 3.
    Lidar D.A.: Towards fault tolerant adiabatic quantum computation. Phys. Rev. Lett. 100, 160506 (2008)CrossRefADSPubMedGoogle Scholar
  4. 4.
    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, 472–475 (2001)CrossRefMathSciNetADSPubMedMATHGoogle Scholar
  5. 5.
    Hogg T.: Adiabatic quantum computing for random satisfiability problems. Phys. Rev. A 67, 022314 (2003)CrossRefADSGoogle Scholar
  6. 6.
    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)CrossRefADSPubMedGoogle Scholar
  7. 7.
    Perdomo A., Truncik C., Tubert-Brohman I., Rose G., Aspuru-Guzik A.: Construction of model Hamiltonians for adiabatic quantum computation and its application to finding low-energy conformations of lattice protein models. Phys. Rev. A 78, 012320 (2008)CrossRefADSGoogle Scholar
  8. 8.
    Farhi, E., Goldstone, J., Gutmann, S.: Quantum adiabatic evolution algorithms with different paths, quant-ph/0208135 (2002)Google Scholar
  9. 9.
    Rezakhani A.T., Kuo W., Hamma A., Lidar D.A., Zanardi P.: Quantum adiabatic brachistochrone. Phys. Rev. Lett. 103, 080502 (2009)CrossRefADSPubMedGoogle Scholar
  10. 10.
    Farhi E., Goldstone J., Gutmann S., Nagaj D.: How to make the quantum adiabatic algorithm fail. Int. J. Quantum Inf. 06, 503–516 (2008)CrossRefGoogle Scholar
  11. 11.
    Roland J., Cerf N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)CrossRefADSGoogle Scholar
  12. 12.
    Znidaric M., Horvat M.: Exponential complexity of an adiabatic algorithm for an NP-complete problem. Phys. Rev. A 73, 022329 (2006)CrossRefADSGoogle Scholar
  13. 13.
    Amin M.H.S.: Effect of local minima on adiabatic quantum optimization. Phys. Rev. Lett. 100, 130503 (2008)CrossRefADSPubMedGoogle Scholar
  14. 14.
    Garey M., Johnson D.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  15. 15.
    Sipser M.: Introduction to the Theory of Computation. PWS Publishing Company, Boston (2005)Google Scholar
  16. 16.
    Hogg T.: Highly structured searches with quantum computers. Phys. Rev. Lett. 80, 2473 (1998)CrossRefADSGoogle Scholar
  17. 17.
    Hogg T.: Quantum search heuristics. Phys. Rev. A 61, 052311 (2000)CrossRefADSGoogle Scholar
  18. 18.
    Grover L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997)CrossRefADSGoogle Scholar
  19. 19.
    Aspuru-Guzik A., Dutoi A.D., Love P.J., Head-Gordon M.: Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005)CrossRefADSPubMedGoogle Scholar
  20. 20.
    Ward N.J., Kassal I., Aspuru-Guzik A.: Preparation of many-body states for quantum simulation. J. Chem. Phys. 130, 194105 (2009)CrossRefADSPubMedGoogle Scholar
  21. 21.
    Wang H., Kais S., Aspuru-Guzik A., Hoffmann M.R.: Quantum algorithm for obtaining the energy spectrum of molecular systems. Phys. Chem. Chem. Phys. 10, 5388–5393 (2008)CrossRefPubMedGoogle Scholar
  22. 22.
    Wang H., Ashhab S., Nori F.: Efficient quantum algorithm for preparing molecular-system-like states on a quantum computer. Phys. Rev. A 79, 042335 (2009)CrossRefADSGoogle Scholar
  23. 23.
    Kohen D., Tannor D.J.: Quantum adiabatic switching. J. Chem. Phys. 98, 3168 (1993)CrossRefADSGoogle Scholar
  24. 24.
    Aharonov, D., Ta-Shma, A.: Adiabatic quantum state generation and statistical zero knowledge, In: Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, pp. 20–29. San Diego, CA, USA, ACM (2003)Google Scholar
  25. 25.
    Messiah A.: Quantum Mechanics (Physics). Dover Publications, Mineola (1999)Google Scholar
  26. 26.
    Marzlin K., Sanders B.C.: Inconsistency in the application of the adiabatic theorem. Phys. Rev. Lett. 93, 160408 (2004)CrossRefADSPubMedGoogle Scholar
  27. 27.
    Tong D.M., Singh K., Kwek L.C., Oh C.H.: Sufficiency criterion for the validity of the adiabatic approximation. Phys. Rev. Lett. 98, 150402 (2007)CrossRefADSPubMedGoogle Scholar
  28. 28.
    Wei Z., Ying M.: Quantum adiabatic computation and adiabatic conditions. Phys. Rev. A 76, 024304 (2007)CrossRefADSGoogle Scholar
  29. 29.
    Zhao Y.: Reexamination of the quantum adiabatic theorem. Phys. Rev. A 77, 032109 (2008)CrossRefADSGoogle Scholar
  30. 30.
    Amin M.H.S.: Consistency of the adiabatic theorem. Phys. Rev. Lett. 102, 220401 (2009)CrossRefMathSciNetADSPubMedGoogle Scholar
  31. 31.
    MacKenzie R., Morin-Duchesne A., Paquette H., Pinel J.: Validity of the adiabatic approximation in quantum mechanics. Phys. Rev. A 76, 044102 (2007)CrossRefADSGoogle Scholar
  32. 32.
    Ambainis, A., Regev, O.: An elementary proof of the quantum adiabatic theorem, quant-ph/0411152s (2004)Google Scholar
  33. 33.
    Jansen S., Ruskai M., Seiler R.: Bounds for the adiabatic approximation with applications to quantum computation. J. Math. Phys. 48, 102111 (2007)CrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Da Wu, J., Sheng Z.M., Lan C.J., De Zhang, Y.: Adiabatic approximation condition, arXiv:0706.0264v2 (2007)Google Scholar
  35. 35.
    Chen, J.-L., Sheng Z.M., Da Wu, J., De Zhang, Y.: Invariant perturbation theory of adiabatic process, http://arxiv.org/abs/0706.0299 (2007)
  36. 36.
    Andrecut M., Ali M.K.: Unstructured adiabatic quantum search. Int. J. Theor. Phys. 43, 925–931 (2004)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Siu M.S.: Adiabatic rotation, quantum search, and preparation of superposition states. Phys. Rev. A 75, 062337 (2007)CrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Acharyya S.: SAT algorithms for colouring some special classes of graphs: some theoretical and experimental results. J. Satisfiability, Boolean Model. Comput. 4, 33–35 (2007)MathSciNetGoogle Scholar
  39. 39.
    Gent, I., Walsh, T.: The search for satisfaction, Internal report, department of computer science, University of Strathclyde (1999)Google Scholar
  40. 40.
    Mezard M., Parisi G., Zecchina R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)CrossRefADSPubMedGoogle Scholar
  41. 41.
    Achlioptas D., Naor A., Peres Y.: Rigorous location of phase transitions in hard optimization problems. Nature 435, 759–764 (2005)CrossRefADSPubMedGoogle Scholar
  42. 42.
    Mezard M., Mora T., Zecchina R.: Clustering of solutions in the random satisfiability problem. Phys. Rev. Lett. 94, 197205 (2005)CrossRefADSPubMedGoogle Scholar
  43. 43.
    Watanabe research group of Department of Mathematics and Computing Sciences, Tokyo Institute of Technology, http://www.is.titech.ac.jp/~watanabe/gensat/
  44. 44.
    Farhi, E., Goldstone, J., Gosset, D., Gutmann, S., Meyer, H., Shor, P.: Quantum adiabatic algorithms, small gaps, and different paths, arxiv.org/abs/0909.4766 (2009)Google Scholar
  45. 45.
    Žnidaric M.: Scaling of the running time of the quantum adiabatic algorithm for propositional satisfiability. Phys. Rev. A 71, 062305 (2005)CrossRefADSGoogle Scholar
  46. 46.
    Du J., Hu L., Wang Y., Wu J., Zhao M., Suter D.: Experimental study of the validity of quantitative conditions in the quantum adiabatic theorem. Phys. Rev. Lett. 101, 060403 (2008)CrossRefADSPubMedGoogle Scholar
  47. 47.
    Kautz H., Selman B.: The state of SAT. Discrete Appl. Math. 155, 1514–1524 (2007)MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Farhi, E., Goldstone, J., Gutmann, S.: Quantum adiabatic evolution algorithms with different paths, quant-ph/0208135 (2002)Google Scholar
  49. 49.
    Kempe J., Kitaev A., Regev O.: The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070–1097 (2006)MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Bravyi, S.: Efficient algorithm for a quantum analogue of 2-SAT, quant-ph/0602108 (2006)Google Scholar
  51. 51.
    Oliveira R., Terhal B.M.: The complexity of quantum system on a two-dimensional square lattice. Quantum Inf. Comput. 8, 0900–0924 (2008)MathSciNetGoogle Scholar
  52. 52.
    Aharonov D., Ta-Shma A.: Adiabatic quantum state generation. SIAM J. Comput. 37, 47–82 (2007)MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Mizel A., Lidar D.A., Mitchell M.: Simple proof of equivalence between adiabatic quantum computation and the circuit model. Phys. Rev. Lett. 99, 070502 (2007)CrossRefADSPubMedGoogle Scholar
  54. 54.
    Schutzhold R., Schaller G.: Adiabatic quantum algorithms as quantum phase transitions: first versus second order. Phys. Rev. A 74, 060304 (2006)CrossRefADSGoogle Scholar
  55. 55.
    Latorre J., Orus R.: Adiabatic quantum computation and quantum phase transitions. Phys. Rev. A 69, 062302 (2004)CrossRefMathSciNetADSGoogle Scholar
  56. 56.
    Orus R., Latorre J.I.: Universality of entanglement and quantum-computation complexity. Phys. Rev. A 69, 052308 (2004)CrossRefMathSciNetADSGoogle Scholar
  57. 57.
    Young, A.P., Knysh, S., Smelyanskiy, V.N.: First-order phase transition in the quantum adiabatic algorithm. Phys. Rev. Lett. 104 (2010)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Alejandro Perdomo-Ortiz
    • 1
  • Salvador E. Venegas-Andraca
    • 2
  • Alán Aspuru-Guzik
    • 1
  1. 1.Department of Chemistry and Chemical BiologyHarvard UniversityCambridgeUSA
  2. 2.Quantum Information Processing GroupTecnológico de Monterrey Campus Estado de MéxicoEdo. MéxicoMéxico

Personalised recommendations