Time-Efficient Quantum Walks for 3-Distinctness

  • Aleksandrs Belovs
  • Andrew M. Childs
  • Stacey Jeffery
  • Robin Kothari
  • Frédéric Magniez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)


We present two quantum walk algorithms for 3-Distinctness. Both algorithms have time complexity \(\tilde{O}(n^{5/7})\), improving the previous \(\tilde{O}(n^{3/4})\) and matching the best known upper bound for query complexity (obtained via learning graphs) up to log factors. The first algorithm is based on a connection between quantum walks and electric networks. The second algorithm uses an extension of the quantum walk search framework that facilitates quantum walks with nested updates.


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  1. 1.
    Buhrman, H., Dürr, C., Heiligman, M., Høyer, P., Santha, M., Magniez, F., de Wolf, R.: Quantum algorithms for Element Distinctness. SIAM Journal on Computing 34, 1324–1330 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambainis, A.: Quantum walk algorithm for element distinctness. In: 45th IEEE FOCS, pp. 22–31 (2004)Google Scholar
  3. 3.
    Aaronson, S., Shi, Y.: Quantum lower bounds for the collision and element distinctness problems. Journal of the ACM 51, 595–605 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Reichardt, B.: Reflections for quantum query algorithms. In: 22nd ACM-SIAM SODA, pp. 560–569 (2011)Google Scholar
  5. 5.
    Lee, T., Mittal, R., Reichardt, B., Spalek, R., Szegedy, M.: Quantum query complexity of state conversion. In: 52nd IEEE FOCS, pp. 344–353 (2011)Google Scholar
  6. 6.
    Belovs, A.: Span programs for functions with constant-sized 1-certificates. In: 44th ACM STOC, pp. 77–84 (2012)Google Scholar
  7. 7.
    Lee, T., Magniez, F., Santha, M.: A learning graph based quantum query algorithm for finding constant-size subgraphs. Chicago Journal of Theoretical Computer Science (2012)Google Scholar
  8. 8.
    Zhu, Y.: Quantum query of subgraph containment with constant-sized certificates. International Journal of Quantum Information 10, 1250019 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lee, T., Magniez, F., Santha, M.: Improved quantum query algorithms for triangle finding and associativity testing. In: ACM-SIAM SODA, pp. 1486–1502 (2013)Google Scholar
  10. 10.
    Belovs, A.: Learning-graph-based quantum algorithm for k-distinctness. In: 53rd IEEE FOCS, pp. 207–216 (2012)Google Scholar
  11. 11.
    Jeffery, S., Kothari, R., Magniez, F.: Nested quantum walks with quantum data structures. In: 24th ACM-SIAM SODA, pp. 1474–1485 (2013)Google Scholar
  12. 12.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. SIAM Journal on Computing 40, 142–164 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer (1998)Google Scholar
  14. 14.
    Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R., Tiwari, P.: The electrical resistance of a graph captures its commute and cover times. Computational Complexity 6, 312–340 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: Proc. of 45th IEEE FOCS, pp. 32–41 (2004)Google Scholar
  16. 16.
    Kitaev, A.: Quantum measurements and the abelian stabilizer problem (1995)Google Scholar
  17. 17.
    Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 454, 339–354 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ambainis, A., Childs, A.M., Reichardt, B.W., Špalek, R., Zhang, S.: Any AND-OR formula of size N can be evaluated in time N 1/2 + o(1) on a quantum computer. SIAM Journal on Computing 39, 2513–2530 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Belovs, A., Lee, T.: Quantum algorithm for k-distinctness with prior knowledge on the input. Technical Report arXiv:1108.3022, arXiv (2011)Google Scholar
  20. 20.
    Kuperberg, G.: Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem (2011)Google Scholar
  21. 21.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proc. of 28th ACM STOC, pp. 212–219 (1996)Google Scholar
  22. 22.
    Childs, A.M., Kothari, R.: Quantum query complexity of minor-closed graph properties. In: 28th STACS, pp. 661–672 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Aleksandrs Belovs
    • 1
  • Andrew M. Childs
    • 2
    • 4
  • Stacey Jeffery
    • 3
    • 4
  • Robin Kothari
    • 3
    • 4
  • Frédéric Magniez
    • 5
  1. 1.Faculty of ComputingUniversity of LatviaLatvia
  2. 2.Department of Combinatorics & OptimizationUniversity of WaterlooCanada
  3. 3.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  4. 4.Institute for Quantum ComputingUniversity of WaterlooCanada
  5. 5.CNRS, LIAFAUniv Paris Diderot, Sorbonne Paris-CitéFrance

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