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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)

Abstract

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.

Keywords

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.

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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

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