Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness

  • Peter Høyer
  • Jan Neerbek
  • Yaoyun Shi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2076)


We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list, we prove a lower bound of 1/π(ln(N) - 1) accesses to the list elements for ordered searching, a lower bound of Ω(N logN) binary comparisons for sorting, and a lower bound of Ω(√N logN) binary comparisons for element distinctness. The previously best known lower bounds are 1/12 log2(N) - O(1) due to Ambainis, Ω(N), and Ω(√N), respectively. Our proofs are based on a weighted all-pairs inner product argument.

In addition to our lower bound results, we give a quantum algorithm for ordered searching using roughly 0:631 log2(N) oracle accesses. Our algorithm uses a quantum routine for traversing through a binary search tree faster than classically, and it is of a nature very different from a faster algorithm due to Farhi, Goldstone, Gutmann, and Sipser.


Quantum Algorithm Query Operator Binary Search Tree Quantum Search Quantum Complexity 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambainis, A.: A better lower bound for quantum algorithms searching an ordered list. Proc. of 40th IEEE FOCS (1999) 352–357Google Scholar
  2. 2.
    Ambainis, A.: Quantum lower bounds by quantum arguments. Proc. of 32nd ACM STOC (2000) 636–643Google Scholar
  3. 3.
    Beals, R., Buhrman, H., Cleve, R., Mosca, M., DE Wolf, R.: Quantum lower bounds by polynomials. Proc. of 39th IEEE FOCS (1998) 352–361Google Scholar
  4. 4.
    Beame, P.: A general sequential time-space tradeoff for finding unique elements. SIAM J. Comput. 20 (1991) 270–277zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bennet, C. H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computation. SIAM J. Comput. 26 (1997) 1510–1523CrossRefMathSciNetGoogle Scholar
  6. 6.
    Borodin, A., Fischer, M. J., Kirkpatrick, D. G., Lynch, NA., Tompa, M.: A time-space tradeoff for sorting on nonoblivious machines. J. Comput. Sys. Sci. 22 (1981) 351–364zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. quant-ph/0005055, 2000Google Scholar
  8. 8.
    Buhrman, H., Dürr, C., Heiligman, M., Høyer, P., Magniez, F., Santha, M., DE Wolf, R.: Quantum algorithms for element distinctness. Proc. of 16th IEEE Computational Complexity (2001) (to appear)Google Scholar
  9. 9.
    Buhrman, H., DE Wolf, R.: A lower bound for quantum search of an ordered list. Inform. Proc. Lett. 70 (1999) 205–209zbMATHCrossRefGoogle Scholar
  10. 10.
    Choi, M.-D.: Tricks or treats with the Hilbert matrix. Amer. Math. Monthly 90 (1983) 301–312zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: A limit on the speed of quantum computation for insertion into an ordered list. quant-ph/9812057, 1998Google Scholar
  12. 12.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Invariant quantum algorithms for insertion into an ordered list. quant-ph/9901059, 1999Google Scholar
  13. 13.
    Grigoriev, D., Karpinski, M., Meyer AUF DER Heide, F., Smolensky, R.: A lower bound for randomized algebraic decision trees. Comput. Complexity 6 (1996/1997) 357–375CrossRefMathSciNetGoogle Scholar
  14. 14.
    Grover, L. K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Letters 79 (1997) 325–328CrossRefGoogle Scholar
  15. 15.
    Jozsa, R., Schlienz, J.: Distinguishability of states and von Neumann entropy. Phys. Rev. A 62 (2000) 012301Google Scholar
  16. 16.
    Shor, P. W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26 (1997) 1484–1509zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Vedral, V.: The role of relative entropy in quantum information theory. quant-ph/0102094, 2001Google Scholar
  18. 18.
    Zalka, Ch.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60 (1999) 2746–2751CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Peter Høyer
    • 1
  • Jan Neerbek
    • 2
  • Yaoyun Shi
    • 3
  1. 1.Dept. of Comp. Sci.University of CalgaryAlbertaCanada
  2. 2.Dept. of Comp. Sci.University of AarhusÅrhus CDenmark
  3. 3.Dept. of Comp. Sci.Princeton UniversityPrincetonUSA

Personalised recommendations