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.
Related problems in quantum computing. In collision problem , we are given a 2-1 function @math and have to find @math such that @math . As shown by Brassard, H yer and Tapp @cite_15 , collision problem can be solved in @math quantum steps instead of @math steps classically. @math is also a quantum lower bound @cite_16 @cite_13 .
@cite_15 @cite_16 @cite_13
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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. 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. 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. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math .