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Quantum Walk Based Search Algorithms

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Theory and Applications of Models of Computation (TAMC 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4978))

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.

Research supported by the European Commission IST Integrated Project Qubit Applications (QAP) 015848, and by the ANR Blanc AlgoQP grant of the French Research Ministry.

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References

  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. Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing 1, 47–79 (2005)

    Article  MathSciNet  Google Scholar 

  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. 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. Ambainis, A.: Quantum walks and their algorithmic applications. International Journal of Quantum Information 1(4), 507–518 (2003)

    Article  MATH  Google Scholar 

  6. Ambainis, A.: Quantum search algorithms. SIGACT News 35(2), 22–35 (2004)

    Article  Google Scholar 

  7. Ambainis, A.: Quantum Walk Algorithm for Element Distinctness. SIAM Journal on Computing 37, 210–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Bennett, C., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM Journal on Computing 26(5), 1510–1523 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press and McGraw-Hill (2001)

    Google Scholar 

  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. 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. Halmos, P.: Finite-dimensional vector spaces. Springer, Heidelberg (1974)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Kempe, J.: Quantum random walks – an introductory survey. Contemporary Physics 44(4), 307–327 (2003)

    Article  MathSciNet  Google Scholar 

  21. Kitaev, A.: Quantum measurements and the abelian stabilizer problem. Electronic Colloquium on Computational Complexity (ECCC) 3 (1996)

    Google Scholar 

  22. Knuth, D.: Sorting and Searching. The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1973)

    Google Scholar 

  23. Magniez, F., Nayak, A.: Quantum complexity of testing group commutativity. Algorithmica 48(3), 221–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Magniez, F., Nayak, A.: Personal communication (2008)

    Google Scholar 

  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. Magniez, F., Santha, M., Szegedy, M.: Quantum Algorithms for the Triangle Problem. SIAM Journal of Computing 37(2), 413–427 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Meyer, D.: From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics 85(5-6), 551–574 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Meyer, D.: On the abscence of homogeneous scalar unitary cellular automata. Physical Letter A 223(5), 337–340 (1996)

    Article  MATH  Google Scholar 

  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. Nayak, A., Vishwanath, A.: Quantum walk on the line. Technical Report quant-ph/0010117, arXiv (2000)

    Google Scholar 

  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. Richter, P.: Almost uniform sampling via quantum walks. New Journal of Physics (to appear)

    Google Scholar 

  33. Schöning, U.: A Probabilistic Algorithm for k -SAT Based on Limited Local Search and Restart. Algorithmica 32(4), 615–623 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Physical Review A 67(052307) (2003)

    Google Scholar 

  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. Vazirani, U.: On the power of quantum computation. Philosophical Transactions of the Royal Society of London, Series A 356, 1759–1768 (1998)

    Article  Google Scholar 

  37. Watrous, J.: Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences 62(2), 376–391 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. CNRS–LRI, Université Paris–Sud, 91405, Orsay, France

    Miklos Santha

  2. Centre for Quantum Technologies, Nat. Univ. of Singapore, Singapore, 117543

    Miklos Santha

Authors
  1. Miklos Santha
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Editor information

Manindra Agrawal Dingzhu Du Zhenhua Duan Angsheng Li

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Santha, M. (2008). Quantum Walk Based Search Algorithms. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2008. Lecture Notes in Computer Science, vol 4978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79228-4_3

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  • DOI: https://doi.org/10.1007/978-3-540-79228-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79227-7

  • Online ISBN: 978-3-540-79228-4

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