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Abstract

In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We present a rather detailed description of a somewhat simplified version of the MNRS algorithm. Finally, in the query complexity model, we show how quantum walks can be applied to the following search problems: Element Distinctness, Matrix Product Verification, Restricted Range Associativity, Triangle, and Group Commutativity.

Keywords

Setup Cost Quantum Algorithm Quantum Walk Quantum Analogue Marked Element 
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.

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References

  1. 1.
    Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and the element distinctness problems. Journal of the ACM, 595–605 (2004)Google Scholar
  2. 2.
    Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing 1, 47–79 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proc. of the 33rd ACM Symposium on Theory of Computing, pp. 50–59 (2001)Google Scholar
  4. 4.
    Aleliunas, R., Karp, R., Lipton, R., Lovász, L., Rackoff, C.: Random Walks, Universal Traversal Sequences, and the cComplexity of Maze Problems. In: Proc. of the 20th Symposium on Foundations of Computer Science, pp. 218–223 (1979)Google Scholar
  5. 5.
    Ambainis, A.: Quantum walks and their algorithmic applications. International Journal of Quantum Information 1(4), 507–518 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ambainis, A.: Quantum search algorithms. SIGACT News 35(2), 22–35 (2004)CrossRefGoogle Scholar
  7. 7.
    Ambainis, A.: Quantum Walk Algorithm for Element Distinctness. SIAM Journal on Computing 37, 210–239 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Ambanis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proc. of the 33rd ACM Symposium on Theory of computing, pp. 37–49 (2001)Google Scholar
  9. 9.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proc. of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 1099–1108 (2005)Google Scholar
  10. 10.
    Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM Journal on Computing 26(5), 1510–1523 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Lomonaco Jr., S.J., Brandt, H.E. (eds.) Quantum Computation and Quantum Information: A Millennium Volume, American Mathematical Society. Contemporary Mathematics Series, vol. 305, pp. 53–74 (2002)Google Scholar
  12. 12.
    Buhrman, H., Spalek, R.: Quantum verification of matrix products. In: Proc. of the 17th ACM-SIAM Symposium on Discrete Algorithms, pp. 880–889 (2006)Google Scholar
  13. 13.
    Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. In: Proc. of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 454(1969), pp. 339–354 (1998)Google Scholar
  14. 14.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press and McGraw-Hill (2001)Google Scholar
  15. 15.
    Dörn, S., Thierauf, T.: The Quantum Query Complexity of Algebraic Properties. In: Proc. of the 16th International Symposium on Fundamentals of Computation Theory, pp. 250–260 (2007)Google Scholar
  16. 16.
    Grover, L.: A fast quantum mechanical algorithm for database search. In: Proc. of the 28th ACM Symposium on the Theory of Computing, pp. 212–219 (1996)Google Scholar
  17. 17.
    Halmos, P.: Finite-dimensional vector spaces. Springer, Heidelberg (1974)zbMATHGoogle Scholar
  18. 18.
    Høyer, P., Mosca, M., de Wolf, R.: Quantum search on bounded-error inputs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 291–299. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Kempe, J.: Discrete Quantum Walks Hit Exponentially Faster. In: Proc. of the International Workshop on Randomization and Approximation Techniques in Computer Science, pp. 354–369 (2003)Google Scholar
  20. 20.
    Kempe, J.: Quantum random walks – an introductory survey. Contemporary Physics 44(4), 307–327 (2003)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Kitaev, A.: Quantum measurements and the abelian stabilizer problem. Electronic Colloquium on Computational Complexity (ECCC) 3 (1996)Google Scholar
  22. 22.
    Knuth, D.: Sorting and Searching. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1973)Google Scholar
  23. 23.
    Magniez, F., Nayak, A.: Quantum complexity of testing group commutativity. Algorithmica 48(3), 221–232 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Magniez, F., Nayak, A.: Personal communication (2008)Google Scholar
  25. 25.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proc. of the 39th ACM Symposium on Theory of Computing, pp. 575–584 (2007)Google Scholar
  26. 26.
    Magniez, F., Santha, M., Szegedy, M.: Quantum Algorithms for the Triangle Problem. SIAM Journal of Computing 37(2), 413–427 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Meyer, D.: From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics 85(5-6), 551–574 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Meyer, D.: On the abscence of homogeneous scalar unitary cellular automata. Physical Letter A 223(5), 337–340 (1996)CrossRefzbMATHGoogle Scholar
  29. 29.
    Moore, C., Russell, A.: Quantum Walks on the Hypercube. In: Proc. of the 6th International Workshop on Randomization and Approximation Techniques in Computer Science, pp. 164–178 (2002)Google Scholar
  30. 30.
    Nayak, A., Vishwanath, A.: Quantum walk on the line. Technical Report quant-ph/0010117, arXiv (2000)Google Scholar
  31. 31.
    Pak, I.: Testing commutativity of a group and the power of randomization (2000), Electronic version http://www-math.mit.edu/~pak/research.html
  32. 32.
    Richter, P.: Almost uniform sampling via quantum walks. New Journal of Physics (to appear)Google Scholar
  33. 33.
    Schöning, U.: A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart. Algorithmica 32(4), 615–623 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Physical Review A 67(052307) (2003)Google Scholar
  35. 35.
    Szegedy, M.: Quantum Speed-Up of Markov Chain Based Algorithms. In: Proc. of the 45th IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)Google Scholar
  36. 36.
    Vazirani, U.: On the power of quantum computation. Philosophical Transactions of the Royal Society of London, Series A 356, 1759–1768 (1998)CrossRefGoogle Scholar
  37. 37.
    Watrous, J.: Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences 62(2), 376–391 (2001)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Miklos Santha
    • 1
    • 2
  1. 1.CNRS–LRIUniversité Paris–SudOrsayFrance
  2. 2.Centre for Quantum TechnologiesNat. Univ. of SingaporeSingapore

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