We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N^ 2 3 ) query quantum algorithm. This improves the previous O(N^ 3 4 ) query quantum algorithm of Buhrman this http URL. (quant-ph 0007016) and matches the lower bound by Shi (quant-ph 0112086). The algorithm also solves the generalization of element distinctness in which we have to find k equal items among N items. For this problem, we get an O(N^ k (k+1) ) query quantum algorithm.
@cite_36: We extend the lower bounds on the depth of algebraic decision trees to the case of randomized algebraic decision trees (with two-sided error) for languages being finite unions of hyperplanes and the intersections of halfspaces. As an application, among other things, we derive, for the first time, @math randomized lower bound for the knapsack problem (which was previously only known for deterministic algebraic decision trees).
In the usual query model (where one query gives one value of @math ), it is easy to see that @math queries are also necessary. Classical lower bounds have also been shown for more general models (e.g. @cite_36 ).