Quantum Information Processing

, Volume 12, Issue 5, pp 2027–2070 | Cite as

Quantum adiabatic machine learning

Article

Abstract

We develop an approach to machine learning and anomaly detection via quantum adiabatic evolution. This approach consists of two quantum phases, with some amount of classical preprocessing to set up the quantum problems. In the training phase we identify an optimal set of weak classifiers, to form a single strong classifier. In the testing phase we adiabatically evolve one or more strong classifiers on a superposition of inputs in order to find certain anomalous elements in the classification space. Both the training and testing phases are executed via quantum adiabatic evolution. All quantum processing is strictly limited to two-qubit interactions so as to ensure physical feasibility. We apply and illustrate this approach in detail to the problem of software verification and validation, with a specific example of the learning phase applied to a problem of interest in flight control systems. Beyond this example, the algorithm can be used to attack a broad class of anomaly detection problems.

Keywords

Adiabatic quantum computation Quantum algorithms  Software verification and validation Anomaly detection 

References

  1. 1.
    Vapnik, V.N.: Statistical Learning Theory. Wiley, London (1998)MATHGoogle Scholar
  2. 2.
    Servedio, R.A., Gortler, S.J.: Equivalences and separations between quantum and classical learnability. SIAM J. Comput. 33, 1067 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aïmeur, E., Brassard, G., Gambs, S.: Machine learning in a quantum world. In: Lamontagne, L., Marchand, M. (eds.) Advances in Artificial Intelligence, vol. 4013 of Lecture Notes in Computer Science, p. 431. Springer, Berlin (2006)Google Scholar
  4. 4.
    Meir, R., Rätsch, G.: An introduction to boosting and leveraging. In: Mendelson, S., Smola, A. (eds.) Advanced Lectures on Machine Learning, vol. 2600 of Lecture Notes in Computer Science, p. 118. Springer, Berlin (2003)Google Scholar
  5. 5.
    Freund, Y., Schapire, R., Abe, N.: A short introduction to boosting. J. Jpn. Soc. Artif. Intell. 14, 771 (1999)Google Scholar
  6. 6.
    Neven, H., Denchev, V.S., Rose, G., Macready, W.G.: Training a binary classifier with the quantum adiabatic algorithm. eprint arXiv:0811.0416Google Scholar
  7. 7.
    Neven, H., Denchev., V.S., Drew-Brook, M., Zhang, J., Macready, W.G., Rose, G.: NIPS 2009 demonstration: Binary classification using hardware implementation of quantum annealing (2009)Google Scholar
  8. 8.
    Chandola, V., Banerjee, A., Kumar, V.: Anomaly detection: A survey. ACM Comput. Surv. (CSUR) 41(3), 15 (2009)CrossRefGoogle Scholar
  9. 9.
    Dijkstra, E.W.: Notes on structured programming. In: Dahl, O.-J., Dijkstra, E.W., Hoare, C.A.R. (eds.) Structured Programming, p. 1. Academic Press, New York (1972)Google Scholar
  10. 10.
    Tassey, G.: The economic impacts of inadequate infrastructure for software testing. National Institute of Standards and Technology, RTI Project 7007.011 (2002)Google Scholar
  11. 11.
    Bryce, R., Kuhn, R., Lei, Y., Kacker, R.: Combinatorial testing. In: Ramachandran, M., de Carvalho, R.A. (eds.) Handbook of Software Engineering Research and Productivity Technologies, p. 196. IGI Global (2009)Google Scholar
  12. 12.
    Kuhn, D.R., Kacker, R.N., Lei, Y.: Practical combinatorial testing. NIST Special, Publication 800–142 (2010)Google Scholar
  13. 13.
    Grindal, M., Offutt, J., Andler, S.F.: Combination Testing Strategies: A survey. GMU Technical, Report ISE-TR-04-05 (2004)Google Scholar
  14. 14.
    Cohen, D.M., Dalal, S.R., Parelius, J., Patton, G.C.: The combinatorial design approach to automatic test generation. Softw. IEEE 13, 83 (1996)CrossRefGoogle Scholar
  15. 15.
    D’Silva, V., Kroening, D., Weissenbacher, G.: A survey of automated techniques for formal software verification. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27, 1165 (2008)CrossRefGoogle Scholar
  16. 16.
    Weber, T., Amjad, H.: Efficiently checking propositional refutations in HOL theorem provers. J. Appl. Log. 7, 26 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Neven, H., Rose, G., Macready, W.G.: Image recognition with an adiabatic quantum computer I. mapping to quadratic unconstrained binary optimization. eprint arXiv:0804.4457Google Scholar
  18. 18.
    Neven, H., Denchev, V.S., Rose, G., Macready, W.G.: Training a large scale classifier with the quantum adiabatic algorithm. eprint arXiv:0912.0779Google Scholar
  19. 19.
    Bian, Z., Chudak, F., Macready, W.G., Rose, G.: The Ising model: teaching an old problem new tricks. D-Wave Systems (2010)Google Scholar
  20. 20.
    Schapire, R.E.: The strength of weak learnability. Mach. Learn. 5, 197 (1990)Google Scholar
  21. 21.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. eprint quant-ph/0001106Google Scholar
  22. 22.
    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(5516), 472 (2001)ADSMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput. 37, 166 (2007)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    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)ADSCrossRefGoogle Scholar
  25. 25.
    Jordan, S.P., Farhi, E., Shor, P.W.: Error-correcting codes for adiabatic quantum computation. Phys. Rev. A 74, 052322 (2006)ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lidar, Daniel A.: Towards fault tolerant adiabatic quantum computation. Phys. Rev. Lett. 100, 160506 (2008)ADSCrossRefGoogle Scholar
  27. 27.
    Childs, Andrew M., Edward, Farhi, John, Preskill: Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (2001)CrossRefGoogle Scholar
  28. 28.
    Sarandy, M.S., Lidar, D.A.: Adiabatic quantum computation in open systems. Phys. Rev. Lett. 95, 250503 (2005)ADSCrossRefGoogle Scholar
  29. 29.
    Stehle, E., Lynch, K., Shevertalov, M., Rorres, C., Mancoridis, S.: On the use of computational geometry to detect software faults at runtime. ICAC10, June 711. Washington, DC, USA (2010)Google Scholar
  30. 30.
    Le Traon, Y., Baudry, B., Jezequel, J.-M.: Design by contract to improve software vigilance. IEEE Trans. Softw. Eng. 32, 571 (2006)CrossRefGoogle Scholar
  31. 31.
    Mannor, S., Meir, R.: Geometric bounds for generalization in boosting. In: Helmbold, D., Williamson, B. (eds.) Computational Learning Theory, vol. 2111 of Lecture Notes in Computer Science, pp. 461–472. Springer, Berlin (2001)Google Scholar
  32. 32.
    Kotsiantis, S.B.: Supervised machine learning: A review of classification techniques. Informatica 31, 249 (2007)MATHMathSciNetGoogle Scholar
  33. 33.
    Yu, L., Liu, H.: Efficient feature selection via analysis of relevance and redundancy. J. Mach. Learn. Res. 5, 1205 (2004)MATHGoogle Scholar
  34. 34.
    Zhang, S., Zhang, C., Yang, Q.: Data preparation for data mining. Appl. Artif. Intell. 17, 375 (2003)CrossRefGoogle Scholar
  35. 35.
    Cheng, H., Yan, X., Han, J., Hsu, C.-W.: Discriminative frequent pattern analysis for effective classification. In: International Conference on Data Engineering, p. 716 (2007)Google Scholar
  36. 36.
    Breiman, L.: Arcing classifiers. Ann. Stat. 26, 801 (1998)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Occam’s razor. Inf. Process. Lett. 24, 377 (1987)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Biamonte, J.D., Peter, Love: Realizable Hamiltonians for universal adiabatic quantum computers. Phys. Rev. A 78, 012352 (2008)ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7, 193 (2008)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Karimi, K., Dickson, N.G., Hamze, F., Amin, M.H.S., Drew-Brook, M., Chudak, F.A., Bunyk, P.I., Macready, W.G., Rose, G.: Investigating the performance of an adiabatic quantum optimization processor. Quantum Inf. Process. 11(1), 77 (2012)Google Scholar
  41. 41.
    Harris, R., Johnson, M.W., Lanting, T., Berkley, A.J., Johansson, J., Bunyk, P., Tolkacheva, E., Ladizinsky, E., Ladizinsky, N., Oh, T., Cioata, F., Perminov, I., Spear, P., Enderud, C., Rich, C., Uchaikin, S., Thom, M.C., Chapple, E.M., Wang, J., Wilson, B., Amin, M.H.S., Dickson, N., Karimi, K., Macready, W., Truncik, C.J.S., Rose, G.: Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor. Phys. Rev. B 82, 024511 (2010)ADSCrossRefGoogle Scholar
  42. 42.
    Cheng, H., Yan, X., Han, J., Hsu, C.-W.: Discriminative frequent pattern analysis for effective classification. In: IEEE 23rd International Conference on Data Engineering, Istanbul, Turkey (2007)Google Scholar
  43. 43.
    Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Springer, Berlin (2003)MATHCrossRefGoogle Scholar
  44. 44.
    Jansen, S., Ruskai, M.-B., Seiler, R.: Bounds for the adiabatic approximation with applications to quantum computation. J. Math. Phys. 48, 102111 (2007)ADSCrossRefMathSciNetGoogle Scholar
  45. 45.
    Lidar, D.A., Rezakhani, A.T., Hamma, A.: Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J. Math. Phys. 50, 102106 (2009)ADSCrossRefMathSciNetGoogle Scholar
  46. 46.
    Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)ADSCrossRefGoogle Scholar
  47. 47.
    Rezakhani, A.T., Pimachev, A.K., Lidar, D.A.: Accuracy versus run time in an adiabatic quantum search. Phys. Rev. A 82, 052305 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    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)ADSCrossRefGoogle Scholar
  49. 49.
    Slepian, D.: On the number of symmetry types of Boolean functions of N variables. Can. J. Math. 5, 185 (1953)MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. C–35, 677 (1986)CrossRefGoogle Scholar
  51. 51.
    Jordan, Stephen P., Edward, Farhi: Perturbative gadgets at arbitrary orders. Phys. Rev. A 77, 062329 (2008)ADSCrossRefGoogle Scholar
  52. 52.
    Rezakhani, A.T., Kuo, W.-J., Hamma, A., Lidar, D.A., Zanardi, P.: Quantum adiabatic brachistochrone. Phys. Rev. Lett. 103, 080502 (2009)ADSCrossRefGoogle Scholar
  53. 53.
    Rezakhani, A.T., Abasto, D.F., Lidar, D.A., Zanardi, P.: Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions. Phys. Rev. A 82, 012321 (2010)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Electrical Engineering, Center for Quantum Information Science and TechnologyUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Departments of Electrical Engineering, Chemistry, and Physics, Center for Quantum Information Science and TechnologyUniversity of Southern CaliforniaLos AngelesUSA

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