License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.8
URN: urn:nbn:de:0030-drops-126774
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12677/
Go to the corresponding LIPIcs Volume Portal


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

pdf-format:
LIPIcs-MFCS-2020-8.pdf (0.6 MB)


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}
}

Keywords: Quantum query complexity, Quantum algorithms, Dyck language, Grid path
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020


DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI