Quantum Complexities of Ordered Searching, Sorting, and Element Distinctness

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 log N) binary comparisons for sorting, and a lower bound of Ω( √ N log N) 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 an exact 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 exact algorithm due to F...

• Type: Conference
• Year: 2001
• Where: ICALP
• Authors: Peter Høyer, Jan Neerbek, Yaoyun Shi
• Keywords: Binary Comparisons, ICALP 2001, Lower Bound, Programming Languages, Quantum Complexities