Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness

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.