, Volume 34, Issue 4, pp 429-448

Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness

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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 $\Omega(\sqrt{N}\log{N})$ 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 $\Omega(\sqrt{N})$ , 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.