Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness
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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.
KeywordsQuantum Algorithm Query Operator Binary Search Tree Quantum Search Quantum Complexity
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