You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
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.

Cited Abstract(s):
[1]: 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 ).

[2]: We show how to search N items arranged on a √N × √N grid in time O(√N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom. since it has been shown recently that such a continuous time walk needs time Ω(N) to perform the same task. Our result improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O(√N) and give several extensions of quantum walk search algorithms and generic expressions for its performance for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another "natural" choice of coin gives a walk that takes Ω(N) steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time O(√N log N).

[3]:
Recent developments. After the work described in this paper, the results and ideas from this paper have been used to construct several other quantum algorithms. [1] have used our element distinctness algorithm to give an @math query quantum algorithm for finding triangles in a graph. [2] have used ideas from the current paper to construct a faster algorithm for search on 2-dimensional grid. Childs and Eisenberg [3] have given a different analysis of our algorithm.