Quantum Lower and Upper Bounds for 2D-Grid and Dyck Language
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.
Quantum query complexity
Quantum algorithms
Dyck language
Grid path
Theory of computation~Quantum query complexity
8:1-8:14
Regular Paper
Supported by QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) and ERDF project 1.1.1.5 A/020 "Quantum algorithms: from complexity theory to experiment". The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities.
A full version of the paper is available at https://arxiv.org/pdf/2007.03402.pdf.
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.
Andris
Ambainis
Andris Ambainis
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Kaspars
Balodis
Kaspars Balodis
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Jānis
Iraids
Jānis Iraids
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Kamil
Khadiev
Kamil Khadiev
Kazan Federal University, Russia
Vladislavs
Kļevickis
Vladislavs Kļevickis
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Krišjānis
Prūsis
Krišjānis Prūsis
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Yixin
Shen
Yixin Shen
Université de Paris, CNRS, IRIF, F-75006 Paris, France
Juris
Smotrovs
Juris Smotrovs
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Jevgēnijs
Vihrovs
Jevgēnijs Vihrovs
Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
10.4230/LIPIcs.MFCS.2020.8
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Andris Ambainis, Kaspars Balodis, Jānis Iraids, Kamil Khadiev, Vladislavs Kļevickis, Krišjānis Prūsis, Yixin Shen, Juris Smotrovs, and Jevgēnijs Vihrovs
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