eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-18
8:1
8:14
10.4230/LIPIcs.MFCS.2020.8
article
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
Ambainis, Andris
1
Balodis, Kaspars
1
Iraids, Jānis
1
Khadiev, Kamil
2
Kļevickis, Vladislavs
1
Prūsis, Krišjānis
1
Shen, Yixin
3
Smotrovs, Juris
1
Vihrovs, Jevgēnijs
1
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Kazan Federal University, Russia
Université de Paris, CNRS, IRIF, F-75006 Paris, France
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 √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 Õ(√n) 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(√n(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 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol170-mfcs2020/LIPIcs.MFCS.2020.8/LIPIcs.MFCS.2020.8.pdf
Quantum query complexity
Quantum algorithms
Dyck language
Grid path