二维网格和Dyck语言的量子上下界
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
摘要：
我们研究了两个问题的量子查询复杂度。首先，我们考虑的问题，确定一个括号的序列是一个适当的平衡（一个戴克字），深度最多$k$。我们称之为$Dyck_{k,n}$问题。我们证明了一个下界的$Ω(c^k \sqrt{n})$，表明这个问题的复杂性呈指数级增加$k$。这里$n$是单词的长度。当$k$是一个常数时，这是有趣的无星语言的一个代表性例子，Aaronson等人最近构造了一个令人惊讶的$\tilde{O}(\sqrt{n})$查询量子算法。他们的证明并没有产生一个通用的算法。当$k$不是常量时，$Dyck_{k,n}$不是上下文无关的。我们给出了一个对所有$k$的$Dyck_{k,n}$进行$O\left(\sqrt{n}(\log{n})^{0.5k}\right)$量子查询的算法.这比$k=o\left(\frac{\log(n)}{\log\log n}\right)$的平凡上限$n$更好。其次，我们考虑在2维网格图的连通性问题，如果一些边缘的网格可能会丢失。通过将“平衡括号”问题嵌入到网格中，我们给出了有向二维网格的$Ω(n^{1.5-ε})$和无向二维网格的$Ω(n^{2-ε})$的下界.有向问题作为一类经典动态规划策略的黑箱模型是有趣的，包括通常用于众所周知的编辑距离问题的策略。我们还展示了一个推广的结果，超过2维。
英文摘要：
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 $Ω(c^k \sqrt{n})$, 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 $\tilde{O}(\sqrt{n})$ query quantum algorithm was recently constructed by Aaronson et al. 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 $O\left(\sqrt{n}(\log{n})^{0.5k}\right)$ quantum queries for $Dyck_{k,n}$ for all $k$. This is better than the trival upper bound $n$ for $k=o\left(\frac{\log(n)}{\log\log n}\right)$. 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 $Ω(n^{1.5-ε})$ for the directed 2D grid and $Ω(n^{2-ε})$ 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.
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