子集寻找的量子算法
Quantum algorithms for subset finding
摘要：
最近，Ambainis给出了一个O（N^（2/3））-查询的量子行走算法来区分元素，更一般地，一个O（N^（L/（L+1）-查询的算法来寻找L个相等的数。我们指出，这个算法实际上解决了一个更一般的问题，找到一个满足任何给定性质的大小为L的子集的问题。我们审查的算法，并给出了一个相当简化的分析其查询复杂性。我们提出了几个应用程序，包括两个算法的问题，找到一个L-团的N-顶点图。其中一个算法使用O（N^（2L/（L+1）边查询，另一个使用\tilde{O}（N^（（5L-2）/（2L+4），这是对L <= 5的改进。后一种算法推广了Magniez，Santha和Szegedy最近的一个结果，他们考虑了L=3的情况（找到一个三角形）。我们还提出了两个开放的问题，连续时间量子行走和下界。
英文摘要：
Recently, Ambainis gave an O(N^(2/3))-query quantum walk algorithm for element distinctness, and more generally, an O(N^(L/(L+1)))-query algorithm for finding L equal numbers. We point out that this algorithm actually solves a much more general problem, the problem of finding a subset of size L that satisfies any given property. We review the algorithm and give a considerably simplified analysis of its query complexity. We present several applications, including two algorithms for the problem of finding an L-clique in an N-vertex graph. One of these algorithms uses O(N^(2L/(L+1))) edge queries, and the other uses \tilde{O}(N^((5L-2)/(2L+4))), which is an improvement for L <= 5. The latter algorithm generalizes a recent result of Magniez, Santha, and Szegedy, who considered the case L=3 (finding a triangle). We also pose two open problems regarding continuous time quantum walk and lower bounds.
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