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.
If element distinctness can be solved with @math queries, then collision problem can be solved with @math queries. (This connection is credited to Andrew Yao in @cite_16 .) Thus, a quantum algorithm for element distinctness implies a quantum algorithm for collision but not the other way around.
@cite_16
2034038586 2949074236 2109371739 2796145042
Given a function f as an oracle, the collision problem is to find two distinct indexes i and j such that f(i) = f(j), under the promise that such indexes exist. Since the security of many fundamental cryptographic primitives depends on the hardness of finding collisions, our lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis. We prove that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function Ω((n r)1 3) times, where n is the size of the domain and r|n. This matches an upper bound of Brassard, Hoyer, and Tapp. No lower bound better than constant was previously known. Our result also implies a quantum lower bound of Ω(n2 3) queries for the element distinctness problem, which is to determine whether n integers are all distinct. The best previous lower bound was Ω(sn) queries. The collision problem is to decide whether a function X: 1,..,n -> 1,..,n is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^ 1 5 ) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O(n^ 1 3 ), but obtaining any lower bound better than Theta(1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound of Theta(n^ 1 7 ) for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory. In this note, we give a quantum algorithm that finds collisions in arbitrary t-to-one functions after only O(3rN t) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover's quantum searching algorithm in a novel way. We present a new quantum algorithm that either finds a triangle (a copy of @math ) in an undirected graph @math on @math nodes, or it outputs reject'' if @math is triangle free. The algorithm uses @math queries, and it is based on a new design concept of Ambainis that incorporates the benefits of quantum walks into Grover search. The algorithm both improves on, and is simpler than a recent algorithm of Szegedy which has @math query complexity. The Triangle Problem was first treated in [BDH+01], where an algorithm with @math query complexity was presented (here @math is the number of edges of @math ).