A general sequential time-space tradeoff for finding unique elements
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Strengths and Weaknesses of Quantum Computing
SIAM Journal on Computing
A lower bound for quantum search of an ordered list
Information Processing Letters
Quantum lower bounds by quantum arguments
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
A Better Lower Bound for Quantum Algorithms Searching an Ordered List
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A Lower Bound for Randomized Algebraic Decision Trees
A Lower Bound for Randomized Algebraic Decision Trees
Quantum Algorithms for Element Distinctness
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
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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(Nlog 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 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.