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标题：Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
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作者：Andris Ambainis ; Kaspars Balodis ; JÄnis Iraids 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：170
页码：8:1-8:14
DOI：10.4230/LIPIcs.MFCS.2020.8
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most k. We call this the Dyck_{k,n} problem. We prove a lower bound of Î©(c^k â^Sn), showing that the complexity of this problem increases exponentially in k. Here n is the length of the word. When k is a constant, this is interesting as a representative example of star-free languages for which a surprising OÌf(â^Sn) query quantum algorithm was recently constructed by Aaronson et al. [Scott Aaronson et al., 2018]. Their proof does not give rise to a general algorithm. When k is not a constant, Dyck_{k,n} is not context-free. We give an algorithm with O(â^Sn(log n)^{0.5k}) quantum queries for Dyck_{k,n} for all k. This is better than the trival upper bound n for k = o({log(n)}/{log log n}). Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the "balanced parentheses" problem into the grid, we show a lower bound of Î©(n^{1.5-Îµ}) for the directed 2D grid and Î©(n^{2-Îµ}) for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.
关键词：Quantum query complexity; Quantum algorithms; Dyck language; Grid path