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DOI: 10.4230/LIPIcs.MFCS.2020.8
URN: urn:nbn:de:0030-drops-126774
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12677/
Ambainis, Andris ;
Balodis, Kaspars ;
Iraids, Jānis ;
Khadiev, Kamil ;
Kļevickis, Vladislavs ;
Prūsis, Krišjānis ;
Shen, Yixin ;
Smotrovs, Juris ;
Vihrovs, Jevgēnijs
Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
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 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.
BibTeX - Entry
@InProceedings{ambainis_et_al:LIPIcs:2020:12677,
author = {Andris Ambainis and Kaspars Balodis and Jānis Iraids and Kamil Khadiev and Vladislavs Kļevickis and Kri{\v{s}}jānis Prūsis and Yixin Shen and Juris Smotrovs and Jevgēnijs Vihrovs},
title = {{Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {8:1--8:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Javier Esparza and Daniel Kr{\'a}ľ},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12677},
URN = {urn:nbn:de:0030-drops-126774},
doi = {10.4230/LIPIcs.MFCS.2020.8},
annote = {Keywords: Quantum query complexity, Quantum algorithms, Dyck language, Grid path}
}
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Keywords: |
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Quantum query complexity, Quantum algorithms, Dyck language, Grid path |
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Collection: |
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45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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18.08.2020 |
