Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language

• URL: http://arxiv.org/abs/2007.03402v2
• Date: Thu, 9 Jul 2020 10:37:53 GMT
• Title: Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
• Authors: Andris Ambainis, Kaspars Balodis, J\=anis Iraids, Kamil Khadiev, Vladislavs K\c{l}evickis, Kri\v{s}j\=anis Pr\=usis, Yixin Shen, Juris Smotrovs, Jevg\=enijs Vihrovs
• Abstract summary: We study the quantum query complexity of two problems. 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 $Omega(n1.5-epsilon)$ for the directed 2D grid and $Omega(n2-epsilon)$ for the undirected 2D grid.
• Score: 1.0118253437732931
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\Omega(c^k \sqrt{n})$, 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 $\Omega(n^{1.5-\epsilon})$ for the directed 2D grid and $\Omega(n^{2-\epsilon})$ 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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