, Volume 34, Issue 4, pp 429-448

First online:

Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness

  •  HoyerAffiliated withDepartment of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N 1N4.
  • ,  NeerbekAffiliated withDepartment of Computer Science, University of Aarhus, DK-8000 Arhus C, Denmark.
  • ,  ShiAffiliated withDepartment of Computer Science, Princeton University, Princeton, NJ 08544, USA.

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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
binary comparisons for element distinctness. The previously best known lower bounds are 1/12 log 2 (N) - O (1) due to Ambainis, Ω(N) , and
, respectively. Our proofs are based on a weighted all-pairs inner product argument.

In addition to our lower bound results, we give an exact quantum algorithm for ordered searching using roughly 0.631 log 2 (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 exact algorithm due to Farhi, Goldstone, Gutmann, and Sipser.

Key words. Quantum computation, Searching, Sorting, Element distinctness, Lower bound.