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Tree search and quantum computation

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Abstract

Traditional tree search algorithms supply a blueprint for modeling problem solving behaviour. A diverse spectrum of problems can be formulated in terms of tree search. Quantum computation, namely Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. In this work we consider the impact of incorporating classical search concepts alongside Grover’s algorithm into a hybrid quantum search system. Some of the crucial points examined include: (1) the reverberations of contemplating the use of non-constant branching factors; (2) determining the consequences of incorporating an heuristic perspective into a quantum tree search model.

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References

  1. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of ACM Symposium on Theory of Computation (STOC’01), pp. 50–59 (2001). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0012090

  2. Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48(2), 1687–1690 (1993). doi:10.1103/PhysRevA.48.1687

    Article  ADS  PubMed  Google Scholar 

  3. Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 1, 507 (2003). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0403120

  4. Ambainis A.: Quantum search algorithms. SIGACT News 35(2), 22–35 (2004). doi:10.1145/992287.992296

    Article  Google Scholar 

  5. Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37, 210 (2007). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0311001

  6. Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: ACM Symposium on Theory of Computing, pp. 37–49 (2001). http://www.citeseer.ist.psu.edu/514019.html

  7. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster (2005). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0402107

  8. Anderson J.R.: The Architecture of Cognition. Harvard University Press, Cambridge (1983)

    Google Scholar 

  9. Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing (1997). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9701001

  10. Boyer, M., Brassard, G., Hoeyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46, 493 (1998). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9605034

  11. Brassard, G., Hoyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation (2000). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0005055

  12. Campbell M., Hoane A.J. Jr., Hsu F.H.: Deep blue. Artif. Intell. 134, 57–83 (2002)

    Article  MATH  Google Scholar 

  13. Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algoritmic speedup by quantum walk. In: Proceedings of the 35th ACM Symposium on Theory of Computing (STOC 2003), pp. 59–68 (2003)

  14. Childs A.M., Farhi E., Gutmann S.: An example of the difference between quantum and classical random walks. Quantum Inf. Process. 1(1), 35–43 (2002)

    Article  MathSciNet  Google Scholar 

  15. Choi, B., Korepin, V.: Quantum Partial Search of a Database with Several Target Items. ArXiv Quantum Phys. e-prints (2006)

  16. Chuang I.L., Gershenfeld N., Kubinec M.: Experimental implementation of fast quantum searching. Phys. Rev. Lett. 80(15), 3408–3411 (1998). doi:10.1103/PhysRevLett.80.3408

    Article  ADS  CAS  Google Scholar 

  17. DeGroot M.H., Schervish M.J.: Probability and Statistics, 3rd edn. Addison-Wesley, London (2002)

    Google Scholar 

  18. Dirac P.A.M.: A new notation for quantum mechanics. Proc. Cambridge Philos. Soc. 35, 416–418 (1939)

    Article  MathSciNet  ADS  Google Scholar 

  19. Dirac P.A.M.: The Principles of Quantum Mechanics, vol. 27 of International series of monographs on physics (Oxford, England), Oxford science publications. Oxford University Press, Oxford (1981)

    Google Scholar 

  20. Edmonds J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ernst G., Newell A., Ernst G.E.: GPS: A Case Study in Generality and Problem Solving. Academic Press, London (1969)

    Google Scholar 

  22. Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58(2), 915–928 (1998). doi:10.1103/PhysRevA.58.915

    Article  ADS  CAS  MathSciNet  Google Scholar 

  23. Feldmann, R.: Game Tree Search on Massively Parallel Systems, Ph.D. thesis. University of Paderborn (1993)

  24. Garey M., Johnson D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences. W.H. Freeman, San Francisco (1979)

    Google Scholar 

  25. Gilchrist W.: Statistical Modelling with Quantile Functions. Chapman and Hall/CRC, London (2000)

    Book  Google Scholar 

  26. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: STOC ’96: Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of computing, pp. 212–219. ACM, New York, NY, USA (1996). doi:10.1145/237814.237866

  27. Grover, L.K.: A framework for fast quantum mechanical algorithms. In: STOC ’98: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 53–62. ACM, New York, NY, USA (1998). doi:10.1145/276698.276712

  28. Grover L.K.: Quantum computers can search rapidly by using almost any transformation. Phys. Rev. Lett. 80(19), 4329–4332 (1998). doi:10.1103/PhysRevLett.80.4329

    Article  ADS  CAS  Google Scholar 

  29. Grover, L.K.: Quantum search on structured problems. Chaos, Solitons & Fractals 10(10), 1695–1705 (1999). doi:10.1016/S0960-0779(98)00217-3. http://www.sciencedirect.com/science/article/B6TJ4-4165DDW-9/2/3163b5ccd7a374c7053fb599a240cf08

  30. Grover L.K.: Trade-offs in the quantum search algorithm. Phys. Rev. A 66(5), 052,314 (2002). doi:10.1103/PhysRevA.66.052314

    Article  CAS  Google Scholar 

  31. Grover L.K.: Fixed-point quantum search. Phys. Rev. Lett. 95(15), 150,501 (2005). doi:10.1103/PhysRevLett.95.150501

    Article  CAS  Google Scholar 

  32. Grover, L.K., Radhakrishnan, J.: Is partial quantum search of a database any easier? (2004). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0407122

  33. Hart P., Nilsson N., Raphael B.: A formal basis for the heuristic determination of minimum cost paths. Syst. Sci. Cybern. IEEE Trans. 4(2), 100–107 (1968). doi:10.1109/TSSC.1968.300136

    Article  Google Scholar 

  34. Hirvensalo M.: Quantum Computing. Springer, Berlin (2004)

    MATH  Google Scholar 

  35. Hogg, T.: A framework for structured quantum search. Phys. D 120, 102 (1998). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9701013

  36. Hopcroft J., Tarjan R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973). doi:10.1145/362248.362272

    Article  Google Scholar 

  37. Hordern E.: Sliding Piece Puzzles. Recreations in Mathematics, No 4. Oxford University Press, USA (1987)

    Google Scholar 

  38. Hsu F.H.: Ibm’s deep blue chess grandmaster chips. Micro. IEEE 19(2), 70–82 (1999)

    Article  Google Scholar 

  39. Hsu F.H.: Behind Deep Blue: Building the Computer That Defeated the World Chess Champion. Princeton University Press, Princeton (2002)

    MATH  Google Scholar 

  40. Hu, C.R.: A family of sure-success quantum algorithms for solving a generalized grover search problem (2002). doi:10.1103/PhysRevA.66.042301

  41. Hughes B.D.: Random Walks and Random Environments, vol 1: Random Walks. Oxford University Press, USA (1995)

    MATH  Google Scholar 

  42. Kaye P.R., Laflamme R., Mosca M.: An Introduction to Quantum Computing. Oxford University Presss, USA (2007)

    MATH  Google Scholar 

  43. Kempe, J.: Quantum random walks—an introductory overview. Contemporary Physics 44, 307 (2003). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0303081

  44. Korepin V., Grover L.: Simple algorithm for partial quantum search. Quantum Information Processing 5, 5–10 (2006). doi:10.1007/s11128-005-0004-z

    Article  MATH  MathSciNet  Google Scholar 

  45. Korepin V.E., Xu Y.: Hierarchical Quantum Search. Int. J. Modern Phys. B 21, 5187–5205 (2007). doi:10.1142/S0217979207038344

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Korf, R.E.: Depth-first iterative deepening: An optimal admissible tree search. Artif. Intell. (1985)

  47. Korf, R.E.: Best-first search with limited memory. UCLA Comput. Sci. Annu. (1991)

  48. Korf R.E.: Linear-space best-first search. Artif. Intell. 62(1), 41–78 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  49. Laird J.E., Newell A., Rosenbloom P.S.: Soar: an architecture for general intelligence. Artif. Intell. 33(1), 1–64 (1987)

    Article  MathSciNet  Google Scholar 

  50. Laird J.E., Rosenbloom P.S., Newell A.: Chunking in soar: the anatomy of a general learning mechanism. Mach. Learn. 1(1), 11–46 (1986)

    Google Scholar 

  51. Long G.L.: Grover algorithm with zero theoretical failure rate. Phys. Rev. A 64(2), 022,307 (2001). doi:10.1103/PhysRevA.64.022307

    Article  CAS  Google Scholar 

  52. Luger G.F., Stubblefield W.A.: Artificial Intelligence: Structures and Strategies for Complex Problem Solving: 2nd edn. The Benjamin/Cummings Publishing Company, Inc, Reading (1993)

    Google Scholar 

  53. Mano M., Kime C.R.: Logic and Computer Design Fundamentals, 2nd edn. Prentice Hall, Englewood Cliffs (2002)

    Google Scholar 

  54. Meyer D.: From quantum cellular automata to quantum lattice gases. J. Stat. Phys. 85(5), 551–574 (1996). doi:10.1007/BF02199356

    Article  ADS  MATH  Google Scholar 

  55. Moore, E.: The shortest path through a maze. In: Proceeding of an International Symposium on the Theory of Switching, Part II, pp. 285–292. Harvard University Press, Cambridge (1959)

  56. Nayak, A., Vishwanath, A.: Quantum walk on the line. Tech. rep., DIMACS Technical Report (2000). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0010117

  57. Newell, A.: A guide to the general problem-solver program gps-2-2. Tech. Rep. RM-3337-PR, RAND Corporation, Santa Monica, CA, USA (1963)

  58. Newell, A., Ernst, G.E.: The search for generality. In: Information Processing 1965: Proceeding of IFIP Congress, vol. 1, pp. 17–24. Spartan, Chicago (1965)

  59. Newell, A., Shaw, J., Simon, H.A.: Report on a general problem-solving program. In: Proceedings of the International Conference on Information Processing, pp. 256–264 (1959)

  60. Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  61. Nilsson N.J.: Principles of Artificial Intelligence. Morgan Kaufmann Publishers, Inc, Los Altos (1982)

    MATH  Google Scholar 

  62. Post, E.: The Two-Valued Iterative Systems of Mathematical Logic. (AM-5). Princeton University Press, Princeton (1941). http://press.princeton.edu/titles/4055.html

  63. Post E.: Formal reductions of the general combinatorial problem. Am. J. Math. 65, 197–268 (1943)

    Article  MATH  MathSciNet  Google Scholar 

  64. Russell S.J., Norvig P., Canny J.F., Edwards D.D., Malik J.M., Thrun S.: Artificial Intelligence: A Modern Approach, 2nd edn. Prentice Hall, Englewood Cliffs (2003)

    Google Scholar 

  65. Santos, A.C., Tarrataca, L., João, C.: An analysis of navigation algorithms for smartphones using j2me. In: In Proceedings of the Second International ICST Conference on MOBILe Wireless MiddleWARE, Operating Systems, and Applications (Mobilware’09, Berlin, Germany, April 28–29), LNICST, vol. 7, pp. 266–279. Springer, Berlin (2009)

  66. Santos, A.C., Tarrataca, L., João, C.: Context inference for mobile applications in the upcase project. In: In Proceedings of the Second International ICST Conference on MOBILe Wireless MiddleWARE, Operating Systems, and Applications (Mobilware’09, Berlin, Germany, April 28–29), LNICST, vol. 7, pp. 352–365. Springer, Berlin (2009)

  67. Shenvi N., Kempe J., Whaley K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67(5), 052,307 (2003). doi:10.1103/PhysRevA.67.052307

    Article  CAS  Google Scholar 

  68. Shor, P.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994). doi:10.1109/SFCS.1994.365700

  69. Slate, D., Atkin, L.R.: Chess 4.5—northwestern university chess program. In: Chess Skill in Man and Machine, pp. 82–118. Springer, Berlin (1977)

  70. Toffoli, T.: Reversible computing. In: Proceedings of the 7th Colloquium on Automata, Languages and Programming, pp. 632–644. Springer, London (1980)

  71. Toffoli, T.: Reversible computing. Tech. rep., Massschusetts Institute of Technology, Laboratory for Computer Science (1980)

  72. Woess W.: Random Walks on Infinite Graphs and Groups. No. 138 in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  73. Zalka, C.: A grover-based quantum search of optimal order for an unknown number of marked elements (1999). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9902049

  74. Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60, 2746 (1999). http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9711070

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Authors and Affiliations

  1. GAIPS/INESC-ID, Lisbon, Portugal

    Luís Tarrataca & Andreas Wichert

  2. Department of Computer Science, Instituto Superior Técnico, Technical University of Lisbon, Avenida Professor Cavaco Silva, 2780-990, Porto Salvo, Portugal

    Luís Tarrataca & Andreas Wichert

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  1. Luís Tarrataca
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  2. Andreas Wichert
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Correspondence to Luís Tarrataca.

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This work was supported by FCT (INESC-ID multiannual funding) through the PIDDAC Program funds and FCT grant DFRH—SFRH/BD/61846/2009.

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Tarrataca, L., Wichert, A. Tree search and quantum computation. Quantum Inf Process 10, 475–500 (2011). https://doi.org/10.1007/s11128-010-0212-z

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