Ambainis, Andris ;
Balodis, Kaspars ;
Iraids, Jānis ;
Khadiev, Kamil ;
Kļevickis, Vladislavs ;
Prūsis, Krišjānis ;
Shen, Yixin ;
Smotrovs, Juris ;
Vihrovs, Jevgēnijs
Quantum Lower and Upper Bounds for 2DGrid 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 starfree 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 contextfree. 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^{1.5ε}) for the directed 2D grid and Ω(n^{2ε}) for the undirected 2D grid. The directed problem is interesting as a blackbox model for a class of classical dynamic programming strategies including the one that is usually used for the wellknown 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 2DGrid and Dyck Language}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {8:18:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771597},
ISSN = {18688969},
year = {2020},
volume = {170},
editor = {Javier Esparza and Daniel Kr{\'a}ľ},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12677},
URN = {urn:nbn:de:0030drops126774},
doi = {10.4230/LIPIcs.MFCS.2020.8},
annote = {Keywords: Quantum query complexity, Quantum algorithms, Dyck language, Grid path}
}
18.08.2020
Keywords: 

Quantum query complexity, Quantum algorithms, Dyck language, Grid path 
Seminar: 

45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Issue date: 

2020 
Date of publication: 

18.08.2020 