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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


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.

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Hoyer, ., Neerbek, . & Shi, . Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness . Algorithmica 34, 429–448 (2002).

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